Abstract

The optical microscope measurement of small objects, 0.5 to 10 μm in diameter, is complicated by the apparent change in the dimension of the object with a change in the spatial coherence of the illumination. Coherent edge-detection methods have been developed for the measurement of line objects on integrated-circuit photo masks and wafers. A generalization is presented of the coherent threshold equation that permits the extension to any state of partial coherence of the illumination as well as extension to the measurement of nonplanar objects. In the latter case, a waveguide model is developed for imaging of lines patterned in thick layers and is compared with experimental data.

© 1982 Optical Society of America

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References

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  1. W. T. Welford, in Optics in Metrology, P. Mollet, ed. (Pergamon, New York, 1960), p. 85.
  2. W. W. Bullis and D. Nyyssonen, in VLSI Electronics: Microstructure Science, Vol. 3, N. G. Einspruch, ed. (Academic, New York, 1982), pp. 301–346.
  3. D. Nyyssonen, “Optical linewidth measurements on wafers,” Proc. Soc. Photo-Opt. Instrum. Eng. 135, 115–119 (1978).
  4. D. Nyyssonen, “Calibration of optical systems for linewidth measurements on wafers,” Proc. Soc. Photo-Opt. Instrum. Eng. 221, 119–126 (1980).
  5. L. C. Martin, The Theory of the Microscope (Blackie, London, 1966), Chaps. V and VIII.
  6. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980),p. 7.
  7. E. C. Kintner, “Method for the calculation of partially coherent imagery,” Appl. Opt. 17, 2747–2753 (1978).
    [Crossref] [PubMed]
  8. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 522–526.
  9. H. H. Hopkins, “Applications of coherence theory in microscopy and interferometry,” J. Opt. Soc. Am. 47, 508–526 (1957).
    [Crossref]
  10. F. Zernike, “The concept of degree of coherence,” Physica 5, 785–795 (1938).
    [Crossref]
  11. B. M. Wastrasiewicz, “Theoretical calculations of straight edges in partially coherent illumination,” Proc. R. Soc. London Ser. A 293, 391–400 (1966).
  12. Ref. 8, p. 490.
  13. Ref. 8, p. 441.
  14. V. Coates, “Computerized optical system for precision linewidth measurements,” in Proceedings of the Microelectronics Measurement Technology Seminar (Benwill, Boston, Mass., 1979), pp. 273–277.
  15. Ref. 8, p. 631.
  16. H. M. Smith, Principles of Holography (Wiley, New York, 1969), Chap. 4.
  17. R. S. Longhurst, Geometrical and Physical Optics, 2nd ed. (Wiley, New York, 1967), pp. 276–278.
  18. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [Crossref]
  19. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [Crossref]
  20. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
    [Crossref]
  21. F. G. Kaspar, “Computation of light transmitted by a thick grating, for application to contact printing,” J. Opt. Soc. Am. 64, 1623–1630 (1974).
    [Crossref]
  22. F. L. McCrackin and J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” Nat. Bur. Stand. (U.S.) Monog. 256, 61–82 (1963).
  23. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), pp. 538–543.
  24. D. Nyyssonen, “High resolution microdensitometry of photographic emulsions,” Ph.D. Thesis (University of Rochester, Rochester, N.Y., 1975).
  25. W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
    [Crossref]
  26. W. Streifer, R. D. Burnham, and D. R. Scifres, “Analysis of grating-coupled GaAs:GaAlAs lasers and waveguides—II: blazing effects,” IEEE J. Quantum Electron. QE-12, 494–499 (1976).
    [Crossref]
  27. W. Lee and W. Streifer, “Radiation loss calculations for corrugated dielectric waveguides,” J. Opt. Soc. Am. 68, 1701–1707 (1978).
    [Crossref]
  28. F. E. Scire and E. C. Teague, “Piezodriven 50-μ m range stage with subnanometer resolution,” Rev. Sci. Instrum. 49, 1735–1740 (1978).
    [Crossref]

1980 (1)

D. Nyyssonen, “Calibration of optical systems for linewidth measurements on wafers,” Proc. Soc. Photo-Opt. Instrum. Eng. 221, 119–126 (1980).

1978 (4)

E. C. Kintner, “Method for the calculation of partially coherent imagery,” Appl. Opt. 17, 2747–2753 (1978).
[Crossref] [PubMed]

D. Nyyssonen, “Optical linewidth measurements on wafers,” Proc. Soc. Photo-Opt. Instrum. Eng. 135, 115–119 (1978).

W. Lee and W. Streifer, “Radiation loss calculations for corrugated dielectric waveguides,” J. Opt. Soc. Am. 68, 1701–1707 (1978).
[Crossref]

F. E. Scire and E. C. Teague, “Piezodriven 50-μ m range stage with subnanometer resolution,” Rev. Sci. Instrum. 49, 1735–1740 (1978).
[Crossref]

1976 (2)

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

W. Streifer, R. D. Burnham, and D. R. Scifres, “Analysis of grating-coupled GaAs:GaAlAs lasers and waveguides—II: blazing effects,” IEEE J. Quantum Electron. QE-12, 494–499 (1976).
[Crossref]

1974 (1)

1973 (1)

1966 (2)

C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
[Crossref]

B. M. Wastrasiewicz, “Theoretical calculations of straight edges in partially coherent illumination,” Proc. R. Soc. London Ser. A 293, 391–400 (1966).

1963 (1)

F. L. McCrackin and J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” Nat. Bur. Stand. (U.S.) Monog. 256, 61–82 (1963).

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

1957 (1)

1938 (1)

F. Zernike, “The concept of degree of coherence,” Physica 5, 785–795 (1938).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 522–526.

Bullis, W. W.

W. W. Bullis and D. Nyyssonen, in VLSI Electronics: Microstructure Science, Vol. 3, N. G. Einspruch, ed. (Academic, New York, 1982), pp. 301–346.

Burckhardt, C. B.

Burnham, R. D.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

W. Streifer, R. D. Burnham, and D. R. Scifres, “Analysis of grating-coupled GaAs:GaAlAs lasers and waveguides—II: blazing effects,” IEEE J. Quantum Electron. QE-12, 494–499 (1976).
[Crossref]

Coates, V.

V. Coates, “Computerized optical system for precision linewidth measurements,” in Proceedings of the Microelectronics Measurement Technology Seminar (Benwill, Boston, Mass., 1979), pp. 273–277.

Colson, J. P.

F. L. McCrackin and J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” Nat. Bur. Stand. (U.S.) Monog. 256, 61–82 (1963).

Felsen, L. B.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), pp. 538–543.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980),p. 7.

Hopkins, H. H.

Kaspar, F. G.

Kintner, E. C.

Lee, W.

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics, 2nd ed. (Wiley, New York, 1967), pp. 276–278.

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), pp. 538–543.

Martin, L. C.

L. C. Martin, The Theory of the Microscope (Blackie, London, 1966), Chaps. V and VIII.

McCrackin, F. L.

F. L. McCrackin and J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” Nat. Bur. Stand. (U.S.) Monog. 256, 61–82 (1963).

Nyyssonen, D.

D. Nyyssonen, “Calibration of optical systems for linewidth measurements on wafers,” Proc. Soc. Photo-Opt. Instrum. Eng. 221, 119–126 (1980).

D. Nyyssonen, “Optical linewidth measurements on wafers,” Proc. Soc. Photo-Opt. Instrum. Eng. 135, 115–119 (1978).

W. W. Bullis and D. Nyyssonen, in VLSI Electronics: Microstructure Science, Vol. 3, N. G. Einspruch, ed. (Academic, New York, 1982), pp. 301–346.

D. Nyyssonen, “High resolution microdensitometry of photographic emulsions,” Ph.D. Thesis (University of Rochester, Rochester, N.Y., 1975).

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980),p. 7.

Scifres, D. R.

W. Streifer, R. D. Burnham, and D. R. Scifres, “Analysis of grating-coupled GaAs:GaAlAs lasers and waveguides—II: blazing effects,” IEEE J. Quantum Electron. QE-12, 494–499 (1976).
[Crossref]

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

Scire, F. E.

F. E. Scire and E. C. Teague, “Piezodriven 50-μ m range stage with subnanometer resolution,” Rev. Sci. Instrum. 49, 1735–1740 (1978).
[Crossref]

Smith, H. M.

H. M. Smith, Principles of Holography (Wiley, New York, 1969), Chap. 4.

Streifer, W.

W. Lee and W. Streifer, “Radiation loss calculations for corrugated dielectric waveguides,” J. Opt. Soc. Am. 68, 1701–1707 (1978).
[Crossref]

W. Streifer, R. D. Burnham, and D. R. Scifres, “Analysis of grating-coupled GaAs:GaAlAs lasers and waveguides—II: blazing effects,” IEEE J. Quantum Electron. QE-12, 494–499 (1976).
[Crossref]

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

Teague, E. C.

F. E. Scire and E. C. Teague, “Piezodriven 50-μ m range stage with subnanometer resolution,” Rev. Sci. Instrum. 49, 1735–1740 (1978).
[Crossref]

Wastrasiewicz, B. M.

B. M. Wastrasiewicz, “Theoretical calculations of straight edges in partially coherent illumination,” Proc. R. Soc. London Ser. A 293, 391–400 (1966).

Welford, W. T.

W. T. Welford, in Optics in Metrology, P. Mollet, ed. (Pergamon, New York, 1960), p. 85.

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 522–526.

Zernike, F.

F. Zernike, “The concept of degree of coherence,” Physica 5, 785–795 (1938).
[Crossref]

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

W. Streifer, D. R. Scifres, and R. D. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides,” IEEE J. Quantum Electron. QE-12, 422–428 (1976).
[Crossref]

W. Streifer, R. D. Burnham, and D. R. Scifres, “Analysis of grating-coupled GaAs:GaAlAs lasers and waveguides—II: blazing effects,” IEEE J. Quantum Electron. QE-12, 494–499 (1976).
[Crossref]

J. Opt. Soc. Am. (5)

Nat. Bur. Stand. (U.S.) Monog. (1)

F. L. McCrackin and J. P. Colson, “Computational techniques for the use of the exact Drude equations in reflection problems,” Nat. Bur. Stand. (U.S.) Monog. 256, 61–82 (1963).

Physica (1)

F. Zernike, “The concept of degree of coherence,” Physica 5, 785–795 (1938).
[Crossref]

Proc. R. Soc. London Ser. A (2)

B. M. Wastrasiewicz, “Theoretical calculations of straight edges in partially coherent illumination,” Proc. R. Soc. London Ser. A 293, 391–400 (1966).

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

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

D. Nyyssonen, “Optical linewidth measurements on wafers,” Proc. Soc. Photo-Opt. Instrum. Eng. 135, 115–119 (1978).

D. Nyyssonen, “Calibration of optical systems for linewidth measurements on wafers,” Proc. Soc. Photo-Opt. Instrum. Eng. 221, 119–126 (1980).

Rev. Sci. Instrum. (1)

F. E. Scire and E. C. Teague, “Piezodriven 50-μ m range stage with subnanometer resolution,” Rev. Sci. Instrum. 49, 1735–1740 (1978).
[Crossref]

Other (13)

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), pp. 538–543.

D. Nyyssonen, “High resolution microdensitometry of photographic emulsions,” Ph.D. Thesis (University of Rochester, Rochester, N.Y., 1975).

L. C. Martin, The Theory of the Microscope (Blackie, London, 1966), Chaps. V and VIII.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980),p. 7.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), pp. 522–526.

W. T. Welford, in Optics in Metrology, P. Mollet, ed. (Pergamon, New York, 1960), p. 85.

W. W. Bullis and D. Nyyssonen, in VLSI Electronics: Microstructure Science, Vol. 3, N. G. Einspruch, ed. (Academic, New York, 1982), pp. 301–346.

Ref. 8, p. 490.

Ref. 8, p. 441.

V. Coates, “Computerized optical system for precision linewidth measurements,” in Proceedings of the Microelectronics Measurement Technology Seminar (Benwill, Boston, Mass., 1979), pp. 273–277.

Ref. 8, p. 631.

H. M. Smith, Principles of Holography (Wiley, New York, 1969), Chap. 4.

R. S. Longhurst, Geometrical and Physical Optics, 2nd ed. (Wiley, New York, 1967), pp. 276–278.

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Figures (19)

Fig. 1
Fig. 1

Parameters for scalar representation of a line object.

Fig. 2
Fig. 2

Plot of limiting values (M → ∞) of the coherent (S = 0) threshold Tc as a function of relative transmittance (or reflectance) T0 for a, ϕ0 = π; b, ϕ0 = π/2; and c, ϕ0 = 0. The threshold corresponding to incoherent illumination (S → ∞) that is insensitive to ϕ0 is shown for comparison (curve d).

Fig. 3
Fig. 3

(a) Same as Fig. 2, except plot is of partially coherent threshold TPC corresponding to S = 1. (b) Same as Fig. 2, except plot is of partially coherent threshold TPC corresponding to S = 2.

Fig. 4
Fig. 4

Variation of TPC with defocus parameter a2 for T0 = 0.1, for different R values: dashed lines, S = ⅕ solid lines, S = ⅔. The coherent edge-detection threshold Tc is indicated in each case and is equivalent to the value of TPC for zero defocus and S = ⅕.

Fig. 5
Fig. 5

Comparison of experimental (—) and theoretical (●) image profiles for a window etched in an approximately 110-nm-thick layer of silicon dioxide on silicon. The calculated curve is based on parameters of T0 = 0.40, ϕ0 = π/2, 0.85 objective N.A., 0.2 condenser N.A., and a wavelength of 530 nm. The experimental system employed a krypton-ion laser at a wavelength of 530.9 nm.

Fig. 6
Fig. 6

Comparison of experimental (—) and theoretical (○) edge profiles for antireflective chromium. The calculated curve is based on parameters of T0 = 0.01,ϕ0 = π/2, 0.9 objective N.A., 0.6 condenser N.A., and a wavelength of 530 nm. The experimental system used a tungsten-halogen lamp with a green filter, which, in combination with the photomultiplier response, produced a spectral response peaked at 530 nm with a bandpass of 60 nm. (Reproduced from Ref. 2.)

Fig. 7
Fig. 7

Parameters for waveguide representation of a line object.

Fig. 8
Fig. 8

Definition of parameters T and ϕ used to characterize line objects. (a) Image profile showing IM and IO, where T0 = IM/IO. When Tc is used for edge detection, the linewidth is given by X2X1. The optical phase difference ϕ is determined from the optical path difference between rays 1 and 2 for (b) transmitted light and (c) reflected light. In each case, the values of ϕ1 and ϕ2 can be calculated from the given complex indices of refraction n ̂ 0 and n ̂ s and the thickness d of the etched layer by using the Fresnel equations. (See Ref. 15.)

Fig. 9
Fig. 9

Relative reflectance R (dashed curve) and phase difference ϕ (solid curve) for silicon dioxide on silicon calculated from the Fresnel equations for varying thickness of silicon dioxide and monochromatic illumination (530 nm) at normal incidence.

Fig. 10
Fig. 10

Image profiles calculated from scalar theory for silicon dioxide on silicon with varying thickness of oxide: a, 90 nm; b, 270 nm; and c, 450 nm. T0 = 0.27 in all cases, and ϕ0 is 0.31π, 0.93π, and 1.54π, respectively.

Fig. 11
Fig. 11

Image profiles calculated from scalar theory for silicon dioxide on silicon with varying thickness of oxide: a, 180 nm; b, 360 nm; and c, 540 nm. T0 = 1.00 in all cases, and ϕ0 is 0.63π, 1.26π, and 1.90π, respectively.

Fig. 12
Fig. 12

Image profiles of an edge object assumed etched in (a) 90-nm- and (b) 600-nm-thick layers of silicon dioxide on silicon as calculated from scalar theory (curve a) and waveguide model (curve b) (wavelength, 530 nm; objective N.A., 0.85; coherence parameter, 0.25).

Fig. 13
Fig. 13

Image profiles calculated from the waveguide model with parameters the same as Fig. 10.

Fig. 14
Fig. 14

Image profiles calculated from the waveguide model with parameters the same as in Fig. 11.

Fig. 15
Fig. 15

Image edge profiles calculated from the waveguide model for a particular pathological case (d = 0.61 μm, W = 3.2 μm for a silicon dioxide line on silicon) in which the slope of the edge image profile does not change appreciably with focus although the linewidth does: a, λ; b, λ/2; c, 0; and d, −λ/2 defocus.

Fig. 16
Fig. 16

Polynomial method of characterizing nonvertical material edges for which the linewidth is described as a function of z.

Fig. 17
Fig. 17

(a) Comparison of experimental (solid curve) and theoretical (●) (based on waveguide model) image profiles for a window etched in a 616-nm-thick layer of silicon dioxide in silicon. The calculated curve is based on n0 = 1.46, n ̂ s = 4.1 + i ( 0.06 ), 0.85 objective N.A., 0.14 condenser N.A., a wavelength of 514 nm, and a linewidth of 4.85 μm. (b) SEM image of oxide line for wafer samples used in (a).

Fig. 18
Fig. 18

Experimental image profile for the same sample as Fig. 17, except the linewidth is smaller. Focus position is near best focus. The secondary minima near the line edge appear to be enhanced because of the smaller linewidth. The exact locations of line edges cannot be determined without comparison with image-profile calculations.

Fig. 19
Fig. 19

Experimental image profile for an etched silicon line with 45-deg edge slope. Etching is to a depth of approximately 1 μm. Lack of symmetry between right and left edges is due to the sample. Exact locations of edges cannot be determined without an appropriate model.

Tables (2)

Tables Icon

Table 1 Convergence of Series in Eq. (17)

Tables Icon

Table 2 Threshold Tc Corresponding to Edge Location for Multiple Line Patterns Near the Resolution Limit of the Imaging Optics

Equations (58)

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T ( x ) = t ( x ) exp [ i ϕ ( x ) ] ,
t ( x ) = { 1 , x 0 T 0 , x > 0
ϕ ( x ) = { 0 , x 0 ϕ 0 , x > 0 .
I ( y ) = C | T ( x ) K ( y x ) d x | 2 ,
I ( 0 ) = | C 0 K ( x ) d x + T 0 exp ( i ϕ 0 ) C 0 K ( x ) d x | 2 .
0 K ( x ) d x = 0 K ( x ) d x = 1 2 K ( x ) d x ,
T c = I ( 0 ) I ( y ) = | 1 2 [ 1 + T 0 exp ( i ϕ 0 ) ] | 2 ,
T c = 0.25 ( 1 + T 0 + 2 T 0 cos ϕ 0 ) ,
T ( x ) = { 1 , 0 | x | W 2 T 0 exp ( i ϕ 0 ) , W 2 < | x | P 2 .
T ( x ) = m c m a m cos ( 2 π m x P ) ,
c 0 = 1 2 [ 1 + T 0 exp ( i ϕ 0 ) ] , c m = 1 2 [ 1 T 0 exp ( i ϕ 0 ) ] m 0 ,
a m = 1 , m = 0 { 2 sin ( π m W P ) π m , m 0 ,
F ( μ ) = K ( x ) exp ( 2 π i μ x ) d x ,
I ( y ) = | [ T ( x ) exp ( 2 π i μ x ) d x ] × F ( μ ) exp ( 2 π i y μ ) d μ | 2 .
I ( y ) = | m c m a m 1 2 [ δ ( m P μ ) + δ ( m P + μ ) ] × F ( μ ) exp ( 2 π i y μ ) d μ | 2 .
I ( y ) = | m = M M c m a m cos ( 2 π m y P ) | 2 .
T c = I ( W 2 ) I ( 0 ) ,
T c = | m = M M c m a m cos ( π m W P ) | 2 | m = M M c m a m | 2 .
a m = sin ( π m 2 ) ( π m 2 ) ,
T c = | 1 2 [ 1 + T 0 exp ( i ϕ 0 ) ] | 2 | 1 2 [ 1 + T 0 exp ( i ϕ 0 ) ] + 1 2 [ 1 T 0 exp ( i ϕ 0 ) ] k = 1 1 / 2 ( M + 1 ) 4 π ( 1 ) k + 1 ( 2 k 1 ) | ( M odd ) .
k = 1 ( 1 ) k + 1 ( 2 k 1 ) = π 4 .
S M = k = 1 1 / 2 ( M + 1 ) 4 π ( 1 ) k + 1 ( 2 k 1 ) , ( M odd ) .
T c = T c 1 4 [ ( 1 + S M ) 2 + T 0 ( 1 S M ) 2 + 2 T 0 ( 1 S M 2 ) cos ϕ ] .
M P λ N.A. objective ,
I ( y ) = n = b n cos ( 2 π n y P ) ,
b n = b n
b n = { A n A n * Ψ ( n P ; 0 ) + n = 1 [ A n + n A n * Ψ ( n + n P ; n P ) + A n n A n * Ψ ( n n P ; n P ) ] } ,
Ψ ( ξ 1 , ξ 2 ) = S ( ξ , η ) × F ( ξ 1 + ξ , η ) F * ( ξ 2 + ξ , η ) d ξ d η ,
T PC = n = b n cos π n W P n = b n
Re ( A n A m * ) = 1 4 ( 1 T 0 ) a n a m , ( ϕ 0 = π / 2 ) ,
F defocus ( μ ) = exp ( i k a 2 μ 2 ) Rec ( μ | M P ) ,
( x ) = n ̂ 2 = n 0 2 SiO 2 , 1 air , { 0 | x | W 2 W 2 < | x | P 2 ,
( x ) = m m cos ( 2 π i m ω 0 x ) .
c 0 = ½ [ 1 + n 0 2 ] , c m = ½ [ 1 n 0 2 ] , m 0 ,
a m = sin ( m π 2 ) ( m π 2 ) = { 2 π ( 1 ) ( m 1 ) / 2 m ( m odd ) 0 ( m even ) .
E y ( x , z ) = m ( D m exp α m z + D m exp α m z ) × l B l m exp [ i k 0 ( s x + λ l ω 0 ) x ] ,
E i = ŷ exp ( i k 0 z ) , H i = x ̂ exp ( i k 0 z ) ,
H x ( x , z ) = m [ D m exp ( α m z ) D m exp ( α m z ) ] × l B l m ( α m ) exp ( i k 0 λ l ω 0 x ) .
E R = ŷ l a l exp { i k 0 [ λ l ω 0 x + s l 1 z ] } ,
H R = l a l [ s l 1 x ̂ + ( λ l ω 0 ) ] × exp { i k 0 [ λ l ω 0 x + s l 1 z ] } s l 1 = [ 1 ( λ l ω 0 ) 2 ] 1 / 2 .
E T = ŷ l γ l exp { i k 0 [ λ l ω 0 x + s l 3 z ] } ,
H T = l γ l [ s l 3 x ̂ + λ l ω 0 ] exp { i k 0 [ λ l ω 0 x + s l 3 z ] } ,
s l 3 [ ( n 3 + i K 3 ) 2 ( λ l ω 0 ) 2 ] 1 / 2
n ̂ s = n 3 + i K 3 .
δ l + a l = m ( D m + D m ) B l m
δ l s l 1 a l = m ( D m D m ) B l m ( α m ) ,
δ l = { 1 , l = 0 0 , l 0 .
m [ D m exp ( α m d ) + D m exp ( α m d ) ] B l m = γ l exp [ i k 0 s l 3 d ]
m [ D m exp ( α m d ) D m exp ( α m d ) ] B l m ( α m ) = s l 3 γ l exp [ i k 0 s l 3 d ] .
m { D m [ S l 1 α m ] + D m [ s l 1 + α m ] } B l m = [ s l 1 1 ] δ l
m { D m [ S l 3 α m ] × exp ( α m d ) + D m [ s l 3 + α m ] exp ( α m d ) } B l m = 0 .
A 1 D A 2 D = R C , A 3 D A 4 D = 0 .
L 1 λ ω 0 .
E R = ŷ l a l exp { i k 0 [ λ l ω 0 x + s l 1 z ] } ,
a l = m ( D m + D m ) B l , m δ l .
I ( y ) = | E R | 2 ,
I ( y ) = | l a l cos ( i k 0 λ l ω 0 x ) | 2 ,
ϕ 0 = π 2