Abstract

The concept of amplitude-wavelength space applied to the development of an x-ray optic provides a means of summarizing and unifying the complex data associated with each stage of its development: the specification; manufacturing tolerances; and metrology. The determination of manufacturing tolerances requires calculation of the image aberrations of perturbed optical systems and is most usefully carried out using a modified asymptotic Debye theory. Its range of applicability and that of other theories is illustrated in amplitude-wavelength space. Subnanometer perturbations are measured by mechanical or optical techniques, and the measuring instruments are calibrated by means of x-ray interferometry in terms of the primary standards of length.

© 1988 Optical Society of America

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References

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  1. M. Stedman, “Metrological Evaluation of Grazing Incidence Mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 316, 2 (1981).
  2. M. Stedman, “Basis for Comparing the Performance of Surface-Measuring Machines,” Precis. Eng. 9, 149 (1987).
    [CrossRef]
  3. M. Stedman, “Mapping the Performance of Surface-Measuring Instruments,” Proc. Soc. Photo-Opt. Instrum. Eng. 803, 138 (1987).
  4. E. C. Teague, T. V. Vorburger, D. Maystre, “Light Scattering from Manufactured Surfaces,” Ann. CIRP 30, No. 2, 563 (1981).
    [CrossRef]
  5. W. H. Carter, “Wave Theory for a Simple Lens,” Opt. Acta 20, 805 (1973).
    [CrossRef]
  6. K. Lindsey, S. T. Smith, C. J. Robbie, “NPL Nanosurf 2: a Sub-Nanometre Accuracy Stylus-Based Surface Texture and Profile Measuring System with a Wide Range and Low Environmental Susceptibility,” CIRP An.37 (1988) in press.
  7. A. E. Ennos, M. S. Virdee, “Precision Measurement of Surface Form by Laser Autocollimation,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 252 (1983).
  8. M. J. Downs, W. H. McGivern, H. J. Ferguson, “Optical System for Measuring the Profiles of Super-Smooth Surfaces,” Precis. Eng. 7, 211 (1985).
    [CrossRef]
  9. A. Franks, “The Metrology of Grazing Incidence Optics at the National Physical Laboratory,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 170 (1986).
  10. K. W. Raine, P. N. Quested, “Interference Film Microscopy for Metal Phase Identification,” Proc. Soc. Photo-Opt. Instrum. Eng. 368, 110 (1983).
  11. K. W. Raine, A. Franks, “Effects of Variations in Refractive Index on the Interferometric Measurement of Microtopography in Glasses,” Opt. Acta 132, 251 (1985).
    [CrossRef]
  12. E. L. Church, S. R. Lange, “Structure Effects in Optical Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 124 (1986).
  13. N. Bobroff, “Residual Errors in Laser Interferometry from Air Turbulence and Nonlinearity,” Appl. Opt. 26, 2676 (1987).
    [CrossRef] [PubMed]
  14. D. K. Bowen, D. G. Chetwynd, S. T. Davies, “Calibration of Surface Roughness Transducers at Angstrom Levels, using X-Ray Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 563, 412 (1985).
  15. R. D. Deslattes, A. Henins, “X-Ray to Visible Wavelength Ratios,” Phys. Rev. Lett. 31, 972 (1973).
    [CrossRef]
  16. P. Debye, “Das Verhalten von Lichtwellen in der Nahe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755 (1909).
    [CrossRef]
  17. R. K. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964).
  18. M. Kline, I. W. Kay, Electromagnetic Theory of Optics (Interscience, New York, 1965).
  19. A. W. Conway, J. L. Synge, The Mathematical Papers of Sir W. R. Hamilton (Cambridge U.P., London, 1931).
  20. F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).
  21. J. A. Kneisly, “Local Curvature of Wavefronts in an Optical System,” J. Opt. Soc. Am. 54, 229 (1964).
    [CrossRef]
  22. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).
  23. W. T. Welford, “Optical Estimation of Statistics of Surface Scattering from Light Scattering Measurements,” Opt. Quantum Electron. 9, 269 (1977).
    [CrossRef]
  24. J. de Goode, P. Mazur, “On the Extinction Theorem in Electrodynamics,” Physica 58, 568 (1972).
    [CrossRef]
  25. D. N. Pattanayak, E. Wolf, “General Form and a New Interpretation of the Ewald Oseen Extinction Theorem,” Opt. Commun. 6, 217 (1972).
    [CrossRef]
  26. J. A. Stratton, L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56, 99 (1939).
    [CrossRef]
  27. G. S. Agarwal, “Interaction of Electromagnetic Waves at Rough Dielectric Surfaces,” Phys. Rev. B 15, 2371 (1977).
    [CrossRef]
  28. M. G. Moharam, T. K. Gaylord, “Diffraction Analysis of Dielectric Surface-Relief Gratings,” J. Opt. Soc. Am. 72, 1385 (1982).
    [CrossRef]

1987 (3)

M. Stedman, “Basis for Comparing the Performance of Surface-Measuring Machines,” Precis. Eng. 9, 149 (1987).
[CrossRef]

M. Stedman, “Mapping the Performance of Surface-Measuring Instruments,” Proc. Soc. Photo-Opt. Instrum. Eng. 803, 138 (1987).

N. Bobroff, “Residual Errors in Laser Interferometry from Air Turbulence and Nonlinearity,” Appl. Opt. 26, 2676 (1987).
[CrossRef] [PubMed]

1986 (2)

A. Franks, “The Metrology of Grazing Incidence Optics at the National Physical Laboratory,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 170 (1986).

E. L. Church, S. R. Lange, “Structure Effects in Optical Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 124 (1986).

1985 (3)

D. K. Bowen, D. G. Chetwynd, S. T. Davies, “Calibration of Surface Roughness Transducers at Angstrom Levels, using X-Ray Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 563, 412 (1985).

M. J. Downs, W. H. McGivern, H. J. Ferguson, “Optical System for Measuring the Profiles of Super-Smooth Surfaces,” Precis. Eng. 7, 211 (1985).
[CrossRef]

K. W. Raine, A. Franks, “Effects of Variations in Refractive Index on the Interferometric Measurement of Microtopography in Glasses,” Opt. Acta 132, 251 (1985).
[CrossRef]

1983 (2)

K. W. Raine, P. N. Quested, “Interference Film Microscopy for Metal Phase Identification,” Proc. Soc. Photo-Opt. Instrum. Eng. 368, 110 (1983).

A. E. Ennos, M. S. Virdee, “Precision Measurement of Surface Form by Laser Autocollimation,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 252 (1983).

1982 (1)

1981 (2)

M. Stedman, “Metrological Evaluation of Grazing Incidence Mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 316, 2 (1981).

E. C. Teague, T. V. Vorburger, D. Maystre, “Light Scattering from Manufactured Surfaces,” Ann. CIRP 30, No. 2, 563 (1981).
[CrossRef]

1977 (2)

W. T. Welford, “Optical Estimation of Statistics of Surface Scattering from Light Scattering Measurements,” Opt. Quantum Electron. 9, 269 (1977).
[CrossRef]

G. S. Agarwal, “Interaction of Electromagnetic Waves at Rough Dielectric Surfaces,” Phys. Rev. B 15, 2371 (1977).
[CrossRef]

1973 (2)

R. D. Deslattes, A. Henins, “X-Ray to Visible Wavelength Ratios,” Phys. Rev. Lett. 31, 972 (1973).
[CrossRef]

W. H. Carter, “Wave Theory for a Simple Lens,” Opt. Acta 20, 805 (1973).
[CrossRef]

1972 (2)

J. de Goode, P. Mazur, “On the Extinction Theorem in Electrodynamics,” Physica 58, 568 (1972).
[CrossRef]

D. N. Pattanayak, E. Wolf, “General Form and a New Interpretation of the Ewald Oseen Extinction Theorem,” Opt. Commun. 6, 217 (1972).
[CrossRef]

1964 (1)

1939 (1)

J. A. Stratton, L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56, 99 (1939).
[CrossRef]

1909 (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nahe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755 (1909).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, “Interaction of Electromagnetic Waves at Rough Dielectric Surfaces,” Phys. Rev. B 15, 2371 (1977).
[CrossRef]

Bass, F. G.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

Bobroff, N.

Bowen, D. K.

D. K. Bowen, D. G. Chetwynd, S. T. Davies, “Calibration of Surface Roughness Transducers at Angstrom Levels, using X-Ray Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 563, 412 (1985).

Carter, W. H.

W. H. Carter, “Wave Theory for a Simple Lens,” Opt. Acta 20, 805 (1973).
[CrossRef]

Chetwynd, D. G.

D. K. Bowen, D. G. Chetwynd, S. T. Davies, “Calibration of Surface Roughness Transducers at Angstrom Levels, using X-Ray Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 563, 412 (1985).

Chu, L. J.

J. A. Stratton, L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56, 99 (1939).
[CrossRef]

Church, E. L.

E. L. Church, S. R. Lange, “Structure Effects in Optical Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 124 (1986).

Conway, A. W.

A. W. Conway, J. L. Synge, The Mathematical Papers of Sir W. R. Hamilton (Cambridge U.P., London, 1931).

Davies, S. T.

D. K. Bowen, D. G. Chetwynd, S. T. Davies, “Calibration of Surface Roughness Transducers at Angstrom Levels, using X-Ray Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 563, 412 (1985).

de Goode, J.

J. de Goode, P. Mazur, “On the Extinction Theorem in Electrodynamics,” Physica 58, 568 (1972).
[CrossRef]

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nahe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755 (1909).
[CrossRef]

Deslattes, R. D.

R. D. Deslattes, A. Henins, “X-Ray to Visible Wavelength Ratios,” Phys. Rev. Lett. 31, 972 (1973).
[CrossRef]

Downs, M. J.

M. J. Downs, W. H. McGivern, H. J. Ferguson, “Optical System for Measuring the Profiles of Super-Smooth Surfaces,” Precis. Eng. 7, 211 (1985).
[CrossRef]

Ennos, A. E.

A. E. Ennos, M. S. Virdee, “Precision Measurement of Surface Form by Laser Autocollimation,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 252 (1983).

Ferguson, H. J.

M. J. Downs, W. H. McGivern, H. J. Ferguson, “Optical System for Measuring the Profiles of Super-Smooth Surfaces,” Precis. Eng. 7, 211 (1985).
[CrossRef]

Franks, A.

A. Franks, “The Metrology of Grazing Incidence Optics at the National Physical Laboratory,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 170 (1986).

K. W. Raine, A. Franks, “Effects of Variations in Refractive Index on the Interferometric Measurement of Microtopography in Glasses,” Opt. Acta 132, 251 (1985).
[CrossRef]

Fuks, I. M.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).

Gaylord, T. K.

Henins, A.

R. D. Deslattes, A. Henins, “X-Ray to Visible Wavelength Ratios,” Phys. Rev. Lett. 31, 972 (1973).
[CrossRef]

Kay, I. W.

M. Kline, I. W. Kay, Electromagnetic Theory of Optics (Interscience, New York, 1965).

Kline, M.

M. Kline, I. W. Kay, Electromagnetic Theory of Optics (Interscience, New York, 1965).

Kneisly, J. A.

Lange, S. R.

E. L. Church, S. R. Lange, “Structure Effects in Optical Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 124 (1986).

Lindsey, K.

K. Lindsey, S. T. Smith, C. J. Robbie, “NPL Nanosurf 2: a Sub-Nanometre Accuracy Stylus-Based Surface Texture and Profile Measuring System with a Wide Range and Low Environmental Susceptibility,” CIRP An.37 (1988) in press.

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964).

Maystre, D.

E. C. Teague, T. V. Vorburger, D. Maystre, “Light Scattering from Manufactured Surfaces,” Ann. CIRP 30, No. 2, 563 (1981).
[CrossRef]

Mazur, P.

J. de Goode, P. Mazur, “On the Extinction Theorem in Electrodynamics,” Physica 58, 568 (1972).
[CrossRef]

McGivern, W. H.

M. J. Downs, W. H. McGivern, H. J. Ferguson, “Optical System for Measuring the Profiles of Super-Smooth Surfaces,” Precis. Eng. 7, 211 (1985).
[CrossRef]

Moharam, M. G.

Pattanayak, D. N.

D. N. Pattanayak, E. Wolf, “General Form and a New Interpretation of the Ewald Oseen Extinction Theorem,” Opt. Commun. 6, 217 (1972).
[CrossRef]

Quested, P. N.

K. W. Raine, P. N. Quested, “Interference Film Microscopy for Metal Phase Identification,” Proc. Soc. Photo-Opt. Instrum. Eng. 368, 110 (1983).

Raine, K. W.

K. W. Raine, A. Franks, “Effects of Variations in Refractive Index on the Interferometric Measurement of Microtopography in Glasses,” Opt. Acta 132, 251 (1985).
[CrossRef]

K. W. Raine, P. N. Quested, “Interference Film Microscopy for Metal Phase Identification,” Proc. Soc. Photo-Opt. Instrum. Eng. 368, 110 (1983).

Robbie, C. J.

K. Lindsey, S. T. Smith, C. J. Robbie, “NPL Nanosurf 2: a Sub-Nanometre Accuracy Stylus-Based Surface Texture and Profile Measuring System with a Wide Range and Low Environmental Susceptibility,” CIRP An.37 (1988) in press.

Smith, S. T.

K. Lindsey, S. T. Smith, C. J. Robbie, “NPL Nanosurf 2: a Sub-Nanometre Accuracy Stylus-Based Surface Texture and Profile Measuring System with a Wide Range and Low Environmental Susceptibility,” CIRP An.37 (1988) in press.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

Stedman, M.

M. Stedman, “Mapping the Performance of Surface-Measuring Instruments,” Proc. Soc. Photo-Opt. Instrum. Eng. 803, 138 (1987).

M. Stedman, “Basis for Comparing the Performance of Surface-Measuring Machines,” Precis. Eng. 9, 149 (1987).
[CrossRef]

M. Stedman, “Metrological Evaluation of Grazing Incidence Mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 316, 2 (1981).

Stratton, J. A.

J. A. Stratton, L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56, 99 (1939).
[CrossRef]

Synge, J. L.

A. W. Conway, J. L. Synge, The Mathematical Papers of Sir W. R. Hamilton (Cambridge U.P., London, 1931).

Teague, E. C.

E. C. Teague, T. V. Vorburger, D. Maystre, “Light Scattering from Manufactured Surfaces,” Ann. CIRP 30, No. 2, 563 (1981).
[CrossRef]

Virdee, M. S.

A. E. Ennos, M. S. Virdee, “Precision Measurement of Surface Form by Laser Autocollimation,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 252 (1983).

Vorburger, T. V.

E. C. Teague, T. V. Vorburger, D. Maystre, “Light Scattering from Manufactured Surfaces,” Ann. CIRP 30, No. 2, 563 (1981).
[CrossRef]

Welford, W. T.

W. T. Welford, “Optical Estimation of Statistics of Surface Scattering from Light Scattering Measurements,” Opt. Quantum Electron. 9, 269 (1977).
[CrossRef]

Wolf, E.

D. N. Pattanayak, E. Wolf, “General Form and a New Interpretation of the Ewald Oseen Extinction Theorem,” Opt. Commun. 6, 217 (1972).
[CrossRef]

Ann. CIRP (1)

E. C. Teague, T. V. Vorburger, D. Maystre, “Light Scattering from Manufactured Surfaces,” Ann. CIRP 30, No. 2, 563 (1981).
[CrossRef]

Ann. Phys. (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nahe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755 (1909).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Opt. Acta (2)

K. W. Raine, A. Franks, “Effects of Variations in Refractive Index on the Interferometric Measurement of Microtopography in Glasses,” Opt. Acta 132, 251 (1985).
[CrossRef]

W. H. Carter, “Wave Theory for a Simple Lens,” Opt. Acta 20, 805 (1973).
[CrossRef]

Opt. Commun. (1)

D. N. Pattanayak, E. Wolf, “General Form and a New Interpretation of the Ewald Oseen Extinction Theorem,” Opt. Commun. 6, 217 (1972).
[CrossRef]

Opt. Quantum Electron. (1)

W. T. Welford, “Optical Estimation of Statistics of Surface Scattering from Light Scattering Measurements,” Opt. Quantum Electron. 9, 269 (1977).
[CrossRef]

Phys. Rev. (1)

J. A. Stratton, L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56, 99 (1939).
[CrossRef]

Phys. Rev. B (1)

G. S. Agarwal, “Interaction of Electromagnetic Waves at Rough Dielectric Surfaces,” Phys. Rev. B 15, 2371 (1977).
[CrossRef]

Phys. Rev. Lett. (1)

R. D. Deslattes, A. Henins, “X-Ray to Visible Wavelength Ratios,” Phys. Rev. Lett. 31, 972 (1973).
[CrossRef]

Physica (1)

J. de Goode, P. Mazur, “On the Extinction Theorem in Electrodynamics,” Physica 58, 568 (1972).
[CrossRef]

Precis. Eng. (2)

M. Stedman, “Basis for Comparing the Performance of Surface-Measuring Machines,” Precis. Eng. 9, 149 (1987).
[CrossRef]

M. J. Downs, W. H. McGivern, H. J. Ferguson, “Optical System for Measuring the Profiles of Super-Smooth Surfaces,” Precis. Eng. 7, 211 (1985).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (7)

A. Franks, “The Metrology of Grazing Incidence Optics at the National Physical Laboratory,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 170 (1986).

K. W. Raine, P. N. Quested, “Interference Film Microscopy for Metal Phase Identification,” Proc. Soc. Photo-Opt. Instrum. Eng. 368, 110 (1983).

M. Stedman, “Metrological Evaluation of Grazing Incidence Mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 316, 2 (1981).

A. E. Ennos, M. S. Virdee, “Precision Measurement of Surface Form by Laser Autocollimation,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 252 (1983).

M. Stedman, “Mapping the Performance of Surface-Measuring Instruments,” Proc. Soc. Photo-Opt. Instrum. Eng. 803, 138 (1987).

E. L. Church, S. R. Lange, “Structure Effects in Optical Surface Metrology,” Proc. Soc. Photo-Opt. Instrum. Eng. 680, 124 (1986).

D. K. Bowen, D. G. Chetwynd, S. T. Davies, “Calibration of Surface Roughness Transducers at Angstrom Levels, using X-Ray Interferometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 563, 412 (1985).

Other (6)

R. K. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, 1964).

M. Kline, I. W. Kay, Electromagnetic Theory of Optics (Interscience, New York, 1965).

A. W. Conway, J. L. Synge, The Mathematical Papers of Sir W. R. Hamilton (Cambridge U.P., London, 1931).

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, Oxford, 1979).

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, Oxford, 1963).

K. Lindsey, S. T. Smith, C. J. Robbie, “NPL Nanosurf 2: a Sub-Nanometre Accuracy Stylus-Based Surface Texture and Profile Measuring System with a Wide Range and Low Environmental Susceptibility,” CIRP An.37 (1988) in press.

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Figures (13)

Fig. 1
Fig. 1

Smax and Rmin plotted in AW space.

Fig. 2
Fig. 2

Specification of a mirror in AW space.

Fig. 3
Fig. 3

Limits of validity in AW space associated with some image calculation theories.

Fig. 4
Fig. 4

Talystep, Nanosurf-2, Ennos-Virdee profilometer, and Downs interferometer in AW space.

Fig. 5
Fig. 5

Zygo 5500 profilometer and Wyko Topo-2D and 3D profilers in AW space.

Fig. 6
Fig. 6

NPL Nanosurf-2.

Fig. 7
Fig. 7

Sensitivity of Nanosurf-2: trace (A) shows surface texture (of polished silica) + instrument noise; trace (B) shows instrument noise only.

Fig. 8
Fig. 8

Successive Nanosurf-2 traces (A) and (B) of a polished glass surface showing reproducibility over a 40-mm scan.

Fig. 9
Fig. 9

Difference curve of traces (A) and (B) of Fig. 8; reproducibility is ~1.5-nm rms.

Fig. 10
Fig. 10

Successive Nanosurf-2 traces A and B of a polished glass surface showing reproducibility of better than 0.25-nm rms over a 5-mm traverse.

Fig. 11
Fig. 11

X-ray interferometer; the 11-mm silicon blades are shown at A.

Fig. 12
Fig. 12

Image of a point source produced by a rough ellipsoid. The intensity profile is the sum of I″(P) (curve A)—the diffuse background due both to the amplitude and correlation length perturbations—and I′(P) (curve B), the unperturbed image reduced by an exponential function of the perturbation amplitude only.

Fig. 13
Fig. 13

Comparison of calculations of the first-order spectrum from a diffraction grating using A, first-order perturbation theory, and B, coupled plane wave theory.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

y = A sin ( 2 π x / W ) .
log A = log ( S max / 2 π ) + log W ,
log A = log ( 1 / 4 π 2 R min ) + 2 log W ,
c ( τ ) = σ 2 exp ( τ 2 / τ 0 2 ) ,
k K 1 / 2 ,
k τ 0 2 sin 3 ϕ / 2 3 σ 1 ,
σ 2 / K τ 0 4 < 1 / 48 ,
k σ η sin ϕ < 1 / 8 and σ 2 η 2 / τ 0 2 < 1 / 8 .
E ( P ) = i k 2 π D E ( s 1 , s 2 ) K s 3 exp [ i k ( W + s i x 1 ) ] d s 1 d s 2 .
r ¯ ( u , υ ) = r ( u , υ ) + η ζ ( u , υ ) n ( u , υ ) ,
W = W ¯ + r = 1 R [ Δ Q · ( s s ) + 2 η ζ s · n ˆ ] ( r ) ,
x ρ σ ρ x 3 σ 3 = W σ ρ
r = 1 R { P 1 ( Δ Q ) P 2 ( ζ 1 , ζ 2 ) } ( r ) .
P 2 ( ζ 1 , ζ 2 ) = 1 2 π σ 2 ( 1 c 2 ) exp [ 1 2 σ 2 ( ζ 1 2 + 2 ζ 1 ζ 2 c ζ 2 2 ) ] ,
c ( ρ ) = c ( ρ ¯ ) = 1 A A ζ ( r ) ζ ( r + τ ) d r .
k τ 0 2 | s · n | 3 / 2 3 σ 1 .
K = K + α H 2 ( L N + L N 2 M M ) + α 2 H ¯ 2 H 2 K ¯ ,
r ( u , υ ) = l 1 ε cos u [ sin u ( cos υ e x + sin υ e y ) + cos u e z ] ,
I ( P ) = | u ( P ) | 2 = I ( P ) + I ( P ) ,
I ( P ) I ( ρ , ψ ) = I ( ρ ) = k 2 64 π 2 | u u ( 1 ε 2 ) V ( u ) 1 2 ε cos u + ε 2 exp ( 2 k 2 η 2 σ 2 cos 2 ϕ ) J 0 ( k ρ sin u ) d u | 2 .
cos ϕ = ( 1 ε cos u ) / ( 1 2 ε cos u + ε 2 ) 1 / 2 .
I ( P ) = I ( ρ ) = k 2 τ 0 2 64 π u u π + π ( 1 ε 2 ) ( 1 ε cos u ) 3 V ( u ) l 2 ( 1 2 ε cos u + ε 2 ) 5 / 2 × F ( α , γ ) sin u d u d υ ,
F ( α , γ ) = exp ( α 2 n = 1 α 2 n n · n ! exp ( γ 2 / n ) ,
α 2 = k 2 η 2 σ 2 cos 2 ψ , γ 2 = 1 4 k 2 τ 0 2 ρ 2 l 2 ( 1 ε cos u ) ( sin 2 υ + cos 2 ϕ cos 2 υ cos 2 u ) .
I s ( ρ ) exp ( 2 k 2 η 2 σ 2 sin 2 ϕ ) I 0 ( P ) ,
I s ( ρ ) A exp ( k 2 ρ 2 τ 0 2 8 f 2 ) ,
R 2 = [ ( 1 + c 2 ) cos ϕ c + cos 2 ϕ ] 2 R 1 2 + ( 2 L c c + cos 2 ϕ ) 2 R 2 2 + ( 2 L c sin ϕ 1 + 2 c cos 2 ϕ + c 2 ) 2 R 3 2 ,
for r D , E e ( r ) + F ( r ) = O ,
for r D + , E s ( r ) F ( r ) = O ,
F ( r ) = 1 k 2 E [ g ( r , r ) n ( r ) · E ( r ) + E ( r ) n ( r ) · g ( r , r ) ] d σ ,
g ( r , r ) = 1 4 π | r r | exp ( i k | r r | ) .
E s ( r ) = P = 0 E ( P ) s ( r ) η P
E ( P ) s = K A ( P ) s ( K ) exp { i [ K + ( k 2 K · K ) 1 / 2 e z ] } d K .
ζ ( r ) = F ( K ) exp ( i K · r ) d K .
A ( 1 ) ( s ) ( K ¯ ) = k 2 ( μ 2 1 ) F ( K K ¯ ) G ( 1 ) .
A ( 2 ) ( s ) ( K ¯ ) = k 2 ( μ 2 1 ) 2 | F ( K K ¯ ) | 2 d K G ( 2 ) ,

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