Abstract

The judgment of the imaging quality of an optical system can be carried out by examining its through-focus intensity distribution. It has been shown in a previous paper that a scalar-wave analysis of the imaging process according to the extended Nijboer–Zernike theory allows the retrieval of the complex pupil function of the imaging system, including aberrations as well as transmission variations. However, the applicability of the scalar analysis is limited to systems with a numerical aperture (NA) value of the order of 0.60 or less; beyond these values polarization effects become significant. In this scalar retrieval method, the complex pupil function is represented by means of the coefficients of its expansion in a series involving the Zernike polynomials. This representation is highly efficient, in terms of number and magnitude of the required coefficients, and lends itself quite well to matching procedures in the focal region. This distinguishes the method from the retrieval schemes in the literature, which are normally not based on Zernike-type expansions, and rather rely on point-by-point matching procedures. In a previous paper [J. Opt. Soc. Am. A 20, 2281 (2003) ] we have incorporated the extended Nijboer–Zernike approach into the Ignatowsky–Richards/Wolf formalism for the vectorial treatment of optical systems with high NA. In the present paper we further develop this approach by defining an appropriate set of functions that describe the energy density distribution in the focal region. Using this more refined analysis, we establish the set of equations that allow the retrieval of aberrations and birefringence from the intensity point-spread function in the focal volume for high-NA systems. It is shown that one needs four analyses of the intensity distribution in the image volume with different states of polarization in the entrance pupil. Only in this way will it be possible to retrieve the “vectorial” pupil function that includes the effects of birefringence induced by the imaging system. A first numerical test example is presented that illustrates the importance of using the vectorial approach and the correct NA value in the aberration retrieval scheme.

© 2005 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).
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  7. J. R. Fienup, J. C. Marron, T. J. Schultz, J. H. Seldin, “Hubble space telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
    [CrossRef] [PubMed]
  8. J. G. Walker, “The phase retrieval problem—a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
    [CrossRef]
  9. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829–832 (1982).
    [CrossRef]
  10. J. R. Fienup, “Phase retrieval for undersampled broadband images,” J. Opt. Soc. Am. A 16, 1831–1837 (1999).
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  11. D. Van Dyck, W. Coene, “A new procedure for wavefunction restoration in high-resolution electron-microscopy,” Optik (Stuttgart) 77, 125–128 (1987).
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    [CrossRef] [PubMed]
  17. J. Wesner, J. Heil, T. Sure, “Reconstructing the pupil function of microscope objectives from the intensity PSF,” in Current Developments in Lens Design and Optical Engineering III, R. E. Fischer, W. J. Smith, and R. B. Johnson, eds., Proc. SPIE4767, 32–43 (2003).
  18. P. Dirksen, J. J.M. Braat, A. J.E.M. Janssen, C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr., Microfabr., Microsyst. 2, 61–68 (2003).
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    [CrossRef]
  21. W. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. 1, 1–36 (1919).
  22. B. Richards, 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]
  23. M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives,” J. Opt. Soc. Am. A 3, 2086–2093 (1986).
    [CrossRef]
  24. M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989).
    [CrossRef]
  25. R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mod. Opt. 40, 337–348 (1993).
    [CrossRef]
  26. R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations,” J. Mod. Opt. 40, 2293–2311 (1993).
    [CrossRef]
  27. C. J.R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–819 (1997).
    [CrossRef]
  28. J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, A. S. van de Nes, “Extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003).
    [CrossRef]
  29. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).
  30. R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman, 1974).
  31. J. P. McGuire, R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations. I. Formulation and example,” J. Opt. Soc. Am. A 7, 1614–1626 (1990).
    [CrossRef]
  32. S.-Y. Lu, R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
    [CrossRef]
  33. G. D. VanWiggeren, R. Roy, “Transmission of linearly polarized light through a single-mode fiber with random fluctuations of birefringence,” Appl. Opt. 38, 3888–3892 (1999).
    [CrossRef]
  34. S. Stallinga, “Axial birefringence in high-numerical-aperture optical systems and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2859 (2001).
    [CrossRef]
  35. S. Stallinga, “Light distribution close to focus in biaxially birefringent media,” J. Opt. Soc. Am. A 21, 1785–1798 (2004).
    [CrossRef]
  36. S. Stallinga, “Strehl ratio for focusing into biaxially birefringent media,” J. Opt. Soc. Am. A 21, 2406–2413 (2004).
    [CrossRef]
  37. C. van der Avoort, J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
    [CrossRef]
  38. P. Dirksen, J. J.M. Braat, A. J.E.M. Janssen, A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the extended Nijboer–Zernike theory,” in Optical Microlithography XVIII, B. W. Smith, ed., Proc. SPIE5754, 262–273 (2005).

2005 (1)

C. van der Avoort, J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

2004 (2)

2003 (3)

2002 (2)

2001 (1)

1999 (2)

1998 (1)

1997 (1)

C. J.R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–819 (1997).
[CrossRef]

1994 (1)

1993 (3)

R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mod. Opt. 40, 337–348 (1993).
[CrossRef]

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations,” J. Mod. Opt. 40, 2293–2311 (1993).
[CrossRef]

J. R. Fienup, J. C. Marron, T. J. Schultz, J. H. Seldin, “Hubble space telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
[CrossRef] [PubMed]

1992 (2)

1990 (1)

1989 (1)

1987 (1)

D. Van Dyck, W. Coene, “A new procedure for wavefunction restoration in high-resolution electron-microscopy,” Optik (Stuttgart) 77, 125–128 (1987).

1986 (1)

1982 (2)

J. R. Fienup, “Phase retrieval algorithms—a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829–832 (1982).
[CrossRef]

1981 (1)

J. G. Walker, “The phase retrieval problem—a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

1978 (1)

1972 (3)

1971 (1)

R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in electron-microscope,” Optik (Stuttgart) 34, 277–286 (1971).

1959 (1)

B. Richards, 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]

1919 (1)

W. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. 1, 1–36 (1919).

Agrad, D. A.

Angel, E. S.

Barakat, R.

Braat, J. J.M.

C. van der Avoort, J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

P. Dirksen, J. J.M. Braat, A. J.E.M. Janssen, C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr., Microfabr., Microsyst. 2, 61–68 (2003).

J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, A. S. van de Nes, “Extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003).
[CrossRef]

J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
[CrossRef]

P. Dirksen, J. J.M. Braat, A. J.E.M. Janssen, A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the extended Nijboer–Zernike theory,” in Optical Microlithography XVIII, B. W. Smith, ed., Proc. SPIE5754, 262–273 (2005).

Chipman, R. A.

Coene, W.

D. Van Dyck, W. Coene, “A new procedure for wavefunction restoration in high-resolution electron-microscopy,” Optik (Stuttgart) 77, 125–128 (1987).

Dirksen, P.

C. van der Avoort, J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

P. Dirksen, J. J.M. Braat, A. J.E.M. Janssen, C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr., Microfabr., Microsyst. 2, 61–68 (2003).

J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, A. S. van de Nes, “Extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003).
[CrossRef]

J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
[CrossRef]

P. Dirksen, J. J.M. Braat, A. J.E.M. Janssen, A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the extended Nijboer–Zernike theory,” in Optical Microlithography XVIII, B. W. Smith, ed., Proc. SPIE5754, 262–273 (2005).

Fienup, J. R.

Frieden, B. R.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for determination of phase from image and diffraction pictures,” Optik (Stuttgart) 35, 237–246 (1972).

R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in electron-microscope,” Optik (Stuttgart) 34, 277–286 (1971).

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829–832 (1982).
[CrossRef]

Gustafsson, M. G.L.

Hanser, B. M.

Heil, J.

J. Wesner, J. Heil, T. Sure, “Reconstructing the pupil function of microscope objectives from the intensity PSF,” in Current Developments in Lens Design and Optical Engineering III, R. E. Fischer, W. J. Smith, and R. B. Johnson, eds., Proc. SPIE4767, 32–43 (2003).

Iglesias, I.

Ignatowsky, W.

W. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. 1, 1–36 (1919).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Jain, A. K.

Janssen, A. J.E.M.

C. van der Avoort, J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

P. Dirksen, J. J.M. Braat, A. J.E.M. Janssen, C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr., Microfabr., Microsyst. 2, 61–68 (2003).

J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, A. S. van de Nes, “Extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003).
[CrossRef]

A. J.E.M. Janssen, “Extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 849–857 (2002).
[CrossRef]

J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
[CrossRef]

P. Dirksen, J. J.M. Braat, A. J.E.M. Janssen, A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the extended Nijboer–Zernike theory,” in Optical Microlithography XVIII, B. W. Smith, ed., Proc. SPIE5754, 262–273 (2005).

Juffermans, C.

P. Dirksen, J. J.M. Braat, A. J.E.M. Janssen, C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr., Microfabr., Microsyst. 2, 61–68 (2003).

Kant, R.

R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mod. Opt. 40, 337–348 (1993).
[CrossRef]

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations,” J. Mod. Opt. 40, 2293–2311 (1993).
[CrossRef]

Leeuwestein, A.

P. Dirksen, J. J.M. Braat, A. J.E.M. Janssen, A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the extended Nijboer–Zernike theory,” in Optical Microlithography XVIII, B. W. Smith, ed., Proc. SPIE5754, 262–273 (2005).

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman, 1974).

Lu, S.-Y.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

Mansuripur, M.

Marron, J. C.

McGuire, J. P.

Oh, C.

Richards, B.

B. Richards, 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]

Richardson, W. H.

Roy, R.

Sandler, B. H.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for determination of phase from image and diffraction pictures,” Optik (Stuttgart) 35, 237–246 (1972).

R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in electron-microscope,” Optik (Stuttgart) 34, 277–286 (1971).

Schultz, T. J.

Sedat, J. W.

Seldin, J. H.

Sheppard, C. J.R.

C. J.R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–819 (1997).
[CrossRef]

Stallinga, S.

Sure, T.

J. Wesner, J. Heil, T. Sure, “Reconstructing the pupil function of microscope objectives from the intensity PSF,” in Current Developments in Lens Design and Optical Engineering III, R. E. Fischer, W. J. Smith, and R. B. Johnson, eds., Proc. SPIE4767, 32–43 (2003).

Török, P.

C. J.R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–819 (1997).
[CrossRef]

van de Nes, A. S.

van der Avoort, C.

C. van der Avoort, J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

Van Dyck, D.

D. Van Dyck, W. Coene, “A new procedure for wavefunction restoration in high-resolution electron-microscopy,” Optik (Stuttgart) 77, 125–128 (1987).

VanWiggeren, G. D.

Walker, J. G.

J. G. Walker, “The phase retrieval problem—a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

Wesner, J.

J. Wesner, J. Heil, T. Sure, “Reconstructing the pupil function of microscope objectives from the intensity PSF,” in Current Developments in Lens Design and Optical Engineering III, R. E. Fischer, W. J. Smith, and R. B. Johnson, eds., Proc. SPIE4767, 32–43 (2003).

Wolf, E.

B. Richards, 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]

Appl. Opt. (6)

J. Microlithogr., Microfabr., Microsyst. (1)

P. Dirksen, J. J.M. Braat, A. J.E.M. Janssen, C. Juffermans, “Aberration retrieval using the extended Nijboer–Zernike approach,” J. Microlithogr., Microfabr., Microsyst. 2, 61–68 (2003).

J. Mod. Opt. (4)

R. Kant, “An analytical solution of vector diffraction for focusing optical systems,” J. Mod. Opt. 40, 337–348 (1993).
[CrossRef]

R. Kant, “An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations,” J. Mod. Opt. 40, 2293–2311 (1993).
[CrossRef]

C. J.R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–819 (1997).
[CrossRef]

C. van der Avoort, J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer–Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (12)

A. J.E.M. Janssen, “Extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 849–857 (2002).
[CrossRef]

J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
[CrossRef]

R. Barakat, B. H. Sandler, “Determination of the wavefront aberration function from measured values of the point-spread function—a 2-dimensional phase retrieval problem (least-squares method),” J. Opt. Soc. Am. A 9, 1715–1723 (1992).
[CrossRef]

J. R. Fienup, “Phase retrieval for undersampled broadband images,” J. Opt. Soc. Am. A 16, 1831–1837 (1999).
[CrossRef]

J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, A. S. van de Nes, “Extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003).
[CrossRef]

S. Stallinga, “Axial birefringence in high-numerical-aperture optical systems and the light distribution close to focus,” J. Opt. Soc. Am. A 18, 2846–2859 (2001).
[CrossRef]

S. Stallinga, “Light distribution close to focus in biaxially birefringent media,” J. Opt. Soc. Am. A 21, 1785–1798 (2004).
[CrossRef]

S. Stallinga, “Strehl ratio for focusing into biaxially birefringent media,” J. Opt. Soc. Am. A 21, 2406–2413 (2004).
[CrossRef]

M. Mansuripur, “Distribution of light at and near the focus of high-numerical-aperture objectives,” J. Opt. Soc. Am. A 3, 2086–2093 (1986).
[CrossRef]

M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989).
[CrossRef]

J. P. McGuire, R. A. Chipman, “Diffraction image formation in optical systems with polarization aberrations. I. Formulation and example,” J. Opt. Soc. Am. A 7, 1614–1626 (1990).
[CrossRef]

S.-Y. Lu, R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
[CrossRef]

Opt. Acta (1)

J. G. Walker, “The phase retrieval problem—a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

Opt. Eng. (Bellingham) (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829–832 (1982).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (3)

R. W. Gerchberg, W. O. Saxton, “Phase determination from image and diffraction plane pictures in electron-microscope,” Optik (Stuttgart) 34, 277–286 (1971).

R. W. Gerchberg, W. O. Saxton, “Practical algorithm for determination of phase from image and diffraction pictures,” Optik (Stuttgart) 35, 237–246 (1972).

D. Van Dyck, W. Coene, “A new procedure for wavefunction restoration in high-resolution electron-microscopy,” Optik (Stuttgart) 77, 125–128 (1987).

Proc. R. Soc. London, Ser. A (1)

B. Richards, 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]

Trans. Opt. Inst. (1)

W. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. 1, 1–36 (1919).

Other (5)

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

J. Wesner, J. Heil, T. Sure, “Reconstructing the pupil function of microscope objectives from the intensity PSF,” in Current Developments in Lens Design and Optical Engineering III, R. E. Fischer, W. J. Smith, and R. B. Johnson, eds., Proc. SPIE4767, 32–43 (2003).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman, 1974).

P. Dirksen, J. J.M. Braat, A. J.E.M. Janssen, A. Leeuwestein, “Aberration retrieval for high-NA optical systems using the extended Nijboer–Zernike theory,” in Optical Microlithography XVIII, B. W. Smith, ed., Proc. SPIE5754, 262–273 (2005).

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Figures (7)

Fig. 1
Fig. 1

Propagation of light in a high-NA optical system. The incident field is specified in the entrance pupil S 0 of the system [polar coordinates ( ρ , θ ) ] with the aid of the amplitude and phase of, e.g., the tangential and radial field vectors ( g 0 and e 0 ) and the unit propagation vector s 0 . After traversal of the optical system, the field vectors and the propagation direction on the exit pupil sphere S 1 [cylindrical co-ordinates ( r , ϕ , z ) ] are specified by, respectively, the vectors g 1 , e 1 , s 1 . The NA of the imaging system is given by NA = n sin α , with n equal to the refractive index of the image space. The nominal image plane position is given by P I . The description of the field vectors according to the scheme in the paper requires a distance R that is rather large so that the aberrations of the system do not significantly influence the directions of the electric field vectors g 1 and e 1 in image space.

Fig. 2
Fig. 2

Functions V 0 , j 0 ( r , f ) (upper row, f = 0 , middle row, f = 2 π ) for the aberration-free case ( NA = 0.95 , linear polarization along the x direction). The horizontal coordinate r is expressed in the diffraction unit λ s 0 with s 0 the numerical aperture of the imaging system. The solid and dotted curves in the first and second rows apply to, respectively, the real and imaginary part of the V 0 , j 0 functions. Lower row: contour plots of the three electric field components E x , E y , E z and of the electric energy density Σ E i 2 . The contour lines for the electric field components have been chosen at 0.5, 0.09, and 0.025; for the electric energy density the levels are 0.75, 0.50, 0.25, 0.017, and 0.005. In the latter contour plot, the dotted circle indicates the circular 0.50 contour of the hypothetical in-focus scalar intensity distribution.

Fig. 3
Fig. 3

Same as Fig. 2 but now with comatic aberration of lowest order ( α 3 1 = 1 ) . The values of the relevant β n , x m coefficients (second-order approximation of the phase aberration function) are β 0 , x 0 = 15 16 , β 2 , x 0 = 1 80 , β 4 , x 0 = 1 16 , β 6 , x 0 = 9 80 , β 3 , x 1 = β 3 , x 1 = i 2 , β 2 , x 2 = β 2 , x 2 = 1 20 , β 6 , x 2 = β 6 , x 2 = 3 40 ; all β n , y m identical zero. In the contour plot of the energy density (lower row, right-hand figure) the contour levels have been chosen at 0.5, 0.1, 0.05, 0.01, 0.005, and 0.002. For comparison, we have also included the dotted contour plot in the center corresponding to the 50% relative height for the hypothetical scalar diffraction image (same comatic aberration value).

Fig. 4
Fig. 4

Eight G k l ( r , f ) functions that contribute to the energy density in the focal volume and that are used in the aberration retrieval scheme. The unit along the axes is the diffraction unit. The contour plots apply to the aberration-free case in the nominal focal plane ( f = 0 , NA = 0.95 ). To visualize the features of the various functions, the contour levels have been changed from plot to plot. G 0 , 0 : 0.75, 0.5, 0.25, 0.10, 0.07, 0.05, 0.02, 0.01, 0.005, and 0.001; Re [ G 0 , 2 ] , Re [ G 0 , 2 ] , and 2 Re [ G 1 , 1 ] : 0.055, 0.015, 0.005, 0.001, 0, 0.001 , 0.005 , 0.015 , 0.055 (contours with negative values are dotted); G 1 , 1 , G 1 , 1 , G 2 , 2 , and G 2 , 2 : 0.12, 0.06, 0.005, 0.002, 0.001, 0.0005. Note that the functions G 1 , 1 , G 1 , 1 , G 2 , 2 , and G 2 , 2 all have a doughnut shape with a zero on axis.

Fig. 5
Fig. 5

Gray-scale plots of the G functions for the aberration-free case in the nominal focal plane. The order of representation is the same as in Fig. 4. The plots of the functions Re [ G 0 , 2 ] , Re [ G 0 , 2 ] , and 2 Re [ G 1 , 1 ] have been coded with gray for zero level and with white and black shades for positive and negative values, respectively. Note the doughnut shape of the functions G 1 , 1 , G 1 , 1 , G 2 , 2 , and G 2 , 2 in the lower row. There is no relationship between the gray levels in the various graphs; all levels are relative with respect to the local maximum or the zero level.

Fig. 6
Fig. 6

Same as Fig. 4 for the upper and middle row. The graphs apply to a system with astigmatic wavefront aberration and the β coefficients (second-order approximation of the aberration function) are given by the following values: β 0 , x 0 = 11 12 , β 2 , x 0 = 1 8 , β 4 , x 0 = 1 24 , β 2 , x 2 = 1 4 ( 3 + i ) , β 2 , x 2 = 1 4 ( 3 + i ) , β 4 , x 4 = 1 16 ( 1 + i 3 ) , β 4 , x 4 = 1 16 ( 1 i 3 ) . In the lower row, contour plots are given for the astigmatic focal distribution with defocus values f of π 3 , 0 and + π 3 , respectively, where the defocus values f = ± π 3 approximately correspond to the image positions of the two focal lines of the astigmatic pencil. The choice of the various contour levels is identical to that in Fig. 4.

Fig. 7
Fig. 7

Variation in the retrieved value of optical aberration coefficients when an incorrect value of the numerical aperture is used and the vectorial imaging effects are not correctly applied. Forward calculation at an NA value of 0.95. Retrieval of the β aberration coefficients (comatic wavefront aberration of 3rd, 5th, and 7th order, respectively, with Zernike coefficients of + 0.1 , 0.02 , and 0.02 radis) at various values of NA. The correct aberration values are retrieved only when the NA value at retrieval is chosen identical to the value used in the forward calculation scheme. The retrieved values for the scalar scheme, + 0.088 , 0.050 , and 0.042 , respectively, are found in the graph at the abscissa value NA = 0 .

Equations (52)

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B x ( ρ , θ ) = A x ( ρ , θ ) exp [ i 2 π W x ( ρ , θ ) ] ,
B y ( ρ , θ ) = A y ( ρ , θ ) exp [ i 2 π W y ( ρ , θ ) ] .
E x ( r , ϕ , f ) = i γ s 0 2 exp [ i f u 0 ] n , m i m β n , x m exp [ i m ϕ ]
× { [ V n , 0 m + s 0 2 2 V n , 2 m exp ( 2 i ϕ ) + s 0 2 2 V n , 2 m exp ( 2 i ϕ ) ] i s 0 2 2 V n , 2 m exp ( 2 i ϕ ) + i s 0 2 2 V n , 2 m exp ( 2 i ϕ ) [ i s 0 V n , 1 m exp ( i ϕ ) + i s 0 V n , 1 m exp ( i ϕ ) ] . }
E y ( r , ϕ , f ) = i γ s 0 2 exp [ i f u 0 ] n , m i m β n , y m exp [ i m ϕ ]
× { [ i s 0 2 2 V n , 2 m exp ( 2 i ϕ ) + i s 0 2 2 V n , 2 m exp ( 2 i ϕ ) V n , 0 m ] s 0 2 2 V n , 2 m exp ( 2 i ϕ ) s 0 2 2 V n , 2 m exp ( 2 i ϕ ) [ s 0 V n , 1 m exp ( i ϕ ) s 0 V n , 1 m exp ( i ϕ ) ] . }
V n , j m = 0 1 ρ j ( 1 + 1 s 0 2 ρ 2 ) j + 1 ( 1 s 0 2 ρ 2 ) 1 4 exp [ i f u 0 ( 1 1 s 0 2 ρ 2 ) ] × R n m ( ρ ) J m + j ( 2 π r ρ ) ρ d ρ .
w e = ϵ 0 4 n r 2 E 2 .
S = ϵ 0 c 2 2 Re [ E × B * ] ,
G k l ( α , β ) = n , m i m exp [ i m ϕ ] α n m V n , k m ( r , f ) exp [ i k ϕ ] × n , m i m exp [ i m ϕ ] β n m * V n , l m * ( r , f ) exp [ i l ϕ ] = n , m , n , m exp [ i ( m m ) π 2 ] exp [ i ( m m + k l ) ϕ ] α n m β n m * V n , k m ( r , f ) V n , l m * ( r , f ) ,
m , m , n , n a m , m ; n , n = n = n 1 n 2 { m = m 1 m 2 a m , m ; n , n + μ = 1 μ max m ( a m , m + μ ; n , n + a m + μ , m ; n , n ) } + ν = 1 ν max { n m ( a m , m ; n , n + ν + a m , m ; n + ν , n ) + μ = 1 μ max n m [ a m , m + μ ; n , n + ν + a m + μ , m ; n , n + ν + a m , m + μ ; n + ν , n + a m + μ , m ; n + ν , n ] } ,
G k l ( α , β ) = exp [ i ( k l ) ϕ ] [ n m α n m β n m * V n , k m V n , l m * + μ = 1 μ max [ exp ( i μ π 2 ) exp ( i μ ϕ ) m n ( α n m β n m + μ * V n , k m V n , l m + μ * ) + exp ( i μ π 2 ) exp ( i μ ϕ ) m n ( α n m + μ β n m * V n , k m + μ V n , l m * ) ] + ν = 1 ν max ( n m ( α n m β n + ν m * V n , k m V n + ν , l m * + α n + ν m β n m * V n + ν , k m V n , l m * ) + μ = 1 μ max { exp ( i μ π 2 ) exp ( i μ ϕ ) [ n m ( α n m β n + ν m + μ * V n , k m V n + ν , l m + μ * + α n + ν m β n m + μ * V n + ν , k m V n , l m + μ * ) ] + exp ( + i μ π 2 ) exp ( + i μ ϕ ) [ n m ( α n m + μ β n + ν m * V n , k m + μ V n + ν , l m * + α n + ν m + μ β n m * V n + ν , k m + μ V n , l m * ) ] } ) ] ,
w e ( r , ϕ , f ) = ϵ 0 n r 2 s 0 4 4 { G 0 , 0 ( β x , β x ) + s 0 2 Re [ G 0 , 2 ( β x , β x i β y ) + G 0 , 2 ( β x , β x + i β y ) ] + s 0 4 4 [ G 2 , 2 ( β x i β y , β x i β y ) + G 2 , 2 ( β x + i β y , β x + i β y ) ] + s 0 4 2 Re [ G 2 , 2 ( β x i β y , β x + i β y ) ] + G 0 , 0 ( β y , β y ) s 0 2 Re [ G 0 , 2 ( β y , i β x + β y ) + G 0 , 2 ( β y , i β x + β y ) ] + s 0 4 4 [ G 2 , 2 ( i β x + β y , i β x + β y ) + G 2 , 2 ( i β x + β y , i β x + β y ) ] + s 0 4 2 Re [ G 2 , 2 ( i β x + β y , i β x + β y ) ] + s 0 2 [ G 1 , 1 ( i β x + β y , i β x + β y ) + G 1 , 1 ( i β x + β y , i β x + β y ) ] + 2 s 0 2 Re [ G 1 , 1 ( i β x + β y , i β x + β y ) ] } ,
G k l ( α 1 + α 2 , β 1 + β 2 ) = G k l ( α 1 , β 1 ) + G k l ( α 1 , β 2 ) + G k l ( α 2 , β 1 ) + G k l ( α 2 , β 2 )
G k l ( α , β ) = G l k * ( β , α ) ,
w e ( r , ϕ , f ) = ϵ 0 n r 2 s 0 4 4 { G 0 , 0 ( β x , β x ) + G 0 , 0 ( β y , β y ) + s 0 2 Re [ G 0 , 2 ( β x , β x ) + i G 0 , 2 ( β x , β y ) + i G 0 , 2 ( β y , β x ) G 0 , 2 ( β y , β y ) ] + s 0 2 Re [ G 0 , 2 ( β x , β x ) i G 0 , 2 ( β x , β y ) i G 0 , 2 ( β y , β x ) G 0 , 2 ( β y , β y ) ] + s 0 4 2 [ G 2 , 2 ( β x , β x ) + i G 2 , 2 ( β x , β y ) i G 2 , 2 ( β y , β x ) + G 2 , 2 ( β y , β y ) ] + s 0 4 2 [ G 2 , 2 ( β x , β x ) i G 2 , 2 ( β x , β y ) + i G 2 , 2 ( β y , β x ) + G 2 , 2 ( β y , β y ) ] + s 0 2 [ G 1 , 1 ( β x , β x ) + i G 1 , 1 ( β x , β y ) i G 1 , 1 ( β y , β x ) + G 1 , 1 ( β y , β y ) ] + s 0 2 [ G 1 , 1 ( β x , β x ) i G 1 , 1 ( β x , β y ) + i G 1 , 1 ( β y , β x ) + G 1 , 1 ( β y , β y ) ] + 2 s 0 2 Re [ G 1 , 1 ( β x , β x ) + i G 1 , 1 ( β x , β y ) + i G 1 , 1 ( β y , β x ) + G 1 , 1 ( β y , β y ) ] } .
G k l ( α , β ) = exp [ i ( k l ) ϕ ] ( α 0 0 β 0 0 * V 0 , k 0 V 0 , l 0 * + μ = 1 μ max [ exp ( i μ π 2 ) exp ( i μ ϕ ) ( α 0 0 β 0 u * V 0 , k 0 V 0 , l μ * + α 0 μ β 0 0 * V 0 , k μ V 0 , l 0 * ) + exp ( + i μ π 2 ) exp ( + i μ ϕ ) ( α 0 0 β 0 μ * V 0 , k 0 V 0 , l μ * + α 0 μ β 0 0 * V 0 , k μ V 0 , l 0 * ) ] + ν = 1 ν max { α 0 0 β ν 0 * V 0 , k 0 V ν , l 0 * + α ν 0 β 0 0 * V ν , k 0 V 0 , l 0 * + μ = 1 μ max [ exp ( i μ π 2 ) exp ( i μ ϕ ) ( α 0 0 β ν μ * V 0 , k 0 V ν , l μ * + α ν μ β 0 0 * V ν , k μ V 0 , l 0 * ) + exp ( + i μ π 2 ) exp ( + i μ ϕ ) ( α 0 0 β ν μ * V 0 , k 0 V ν , l μ * + α ν μ β 0 0 * V ν , k μ V 0 , l 0 * ) ] } ) .
G k l ( α , β ) = exp [ i ( k l ) ϕ ] ν = 0 ν max μ = μ max ( ν ) + μ max ( ν ) { exp [ i μ π 2 ] exp [ i μ ϕ ] α 0 0 β ν μ * V 0 , k 0 V ν , l μ * + ( 1 ϵ ν μ ) exp [ i μ π 2 ] exp [ i μ ϕ ] α ν μ β 0 0 * V ν , k μ V 0 , l 0 * } ,
G k k ( α , α ) = α 0 0 2 V 0 , k 0 2 + 2 ν = 0 ν max μ = μ max ( ν ) + μ max ( ν ) Re { exp [ i μ π 2 ] exp [ i μ ϕ ] α ν μ α 0 0 * V ν , k μ V 0 , k 0 * } ,
β n , x m = a β n m , β n , y m = b β n m ,
w e ( r , ϕ , f ) 0 = ϵ 0 n r 2 s 0 4 4 ( G 0 , 0 ( β , β ) + s 0 2 { [ a 2 b 2 ] Re [ G 0 , 2 ( β , β ) ] 2 Re ( a b * ) Im [ G 0 , 2 ( β , β ) ] } + s 0 2 { [ a 2 b 2 ] Re [ G 0 , 2 ( β , β ) ] + 2 Re ( a b * ) Im [ G 0 , 2 ( β , β ) ] } + s 0 4 2 { [ 1 2 Im ( a b * ) ] G 2 , 2 ( β , β ) + [ 1 + 2 Im ( a b * ) ] G 2 , 2 ( β , β ) } + s 0 2 { [ 1 2 Im ( a b * ) ] G 1 , 1 ( β , β ) + [ 1 + 2 Im ( a b * ) ] G 1 , 1 ( β , β ) } 2 s 0 2 { [ a 2 b 2 ] Re [ G + 1 , 1 ( β , β ) ] + 2 Re ( a b * ) Im [ G + 1 , 1 ( β , β ) ] } ) .
w e x ( r , ϕ , f ) 0 G 0 , 0 ( β , β ) + s 0 2 Re [ G 0 , 2 ( β , β ) + G 0 , 2 ( β , β ) ] + s 0 4 2 [ G 2 , 2 ( β , β ) + G 2 , 2 ( β , β ) ] + s 0 2 [ G 1 , 1 ( β , β ) + G 1 , 1 ( β , β ) ] 2 s 0 2 Re [ G + 1 , 1 ( β , β ) ] .
w e y ( r , ϕ , f ) 0 G 0 , 0 ( β , β ) s 0 2 Re [ G 0 , 2 ( β , β ) + G 0 , 2 ( β , β ) ] + s 0 4 2 [ G 2 , 2 ( β , β ) + G 2 , 2 ( β , β ) ] + s 0 2 [ G 1 , 1 ( β , β ) + G 1 , 1 ( β , β ) ] + 2 s 0 2 Re [ G + 1 , 1 ( β , β ) ] .
Δ w l , 0 = w e x ( r , ϕ , f ) 0 w e y ( r , ϕ , f ) 0 = 2 s 0 2 Re [ G 0 , 2 ( β , β ) + G 0 , 2 ( β , β ) 2 G + 1 , 1 ( β , β ) ] .
Δ w l , π 4 = w e x ( r , ϕ , f ) 3 π 4 w e y ( r , ϕ , f ) π 4 = 2 s 0 2 Im [ G 0 , 2 ( β , β ) G 0 , 2 ( β , β ) + 2 G + 1 , 1 ( β , β ) ] .
w e RC ( r , ϕ , f ) 0 G 0 , 0 ( β , β ) + s 0 4 G 2 , 2 ( β , β ) + 2 s 0 2 G 1 , 1 ( β , β ) ,
w e LC ( r , ϕ , f ) 0 G 0 , 0 ( β , β ) + s 0 4 G 2 , 2 ( β , β ) + 2 s 0 2 G 1 , 1 ( β , β ) .
Δ w C , 0 = s 0 4 [ G 2 , 2 ( β , β ) G 2 , 2 ( β , β ) ] + 2 s 0 2 [ G 1 , 1 ( β , β ) G 1 , 1 ( β , β ) ] .
w e N ( r , ϕ , f ) = G 0 , 0 ( β , β ) + s 0 2 [ G 1 , 1 ( β , β ) + G 1 , 1 ( β , β ) ] + s 0 4 2 [ G 2 , 2 ( β , β ) + G 2 , 2 ( β , β ) ] .
( E x E y ) = [ m 11 m 12 m 21 m 22 ] ( a j b j ) ,
M = [ m 11 m 12 m 12 * m 11 * ] ,
Ψ m ( r , f ) = 1 2 π π + π I ( r , ϕ , f ) exp ( i m ϕ ) d ϕ ,
w a n ( r , ϕ , f ) = k , l F k , l ( r , ϕ , f ) ,
F 0 , 0 = G 0 , 0 ( β , β ) ,
F 0 , 2 = s 0 2 { ( a 2 b 2 ) Re [ G 0 , 2 ( β , β ) ] 2 Re ( a b * ) Im [ G 0 , 2 ( β , β ) ] } ,
F 0 , 2 = s 0 2 { [ a 2 b 2 ] Re [ G 0 , 2 ( β , β ) ] + 2 Re ( a b * ) Im [ G 0 , 2 ( β , β ) ] } ,
F + 1 , 1 = 2 s 0 2 { ( a 2 b 2 ) Re [ G + 1 , 1 ( β , β ) ] + 2 Re ( a b * ) Im [ G + 1 , 1 ( β , β ) ] } ,
F 1 , 1 = s 0 2 [ 1 2 Im ( a b * ) ] G 1 , 1 ( β , β ) ,
F 1 , 1 = s 0 2 [ 1 + 2 Im ( a b * ) ] G 1 , 1 ( β , β ) ,
F 2 , 2 = s 0 4 2 [ 1 2 Im ( a b * ) ] G 2 , 2 ( β , β ) ,
F 2 , 2 = s 0 4 2 [ 1 + 2 Im ( a b * ) ] G 2 , 2 ( β , β ) .
G k , l ( β , β ) } = β 0 0 exp [ i ( k l ) ϕ ] ν μ [ β ν μ * Ψ ν ; k , l μ * ( r , f ) exp ( i μ ϕ ) + ( 1 ϵ ν μ ) β ν μ Ψ ν ; l , k μ ( r , f ) exp ( + i μ ϕ ) ] ,
Ψ ν ; k , l μ ( r , f ) = ( + i ) μ V 0 , k 0 * ( r , f ) V ν , l μ ( r , f ) .
1 2 π π + π G k , l ( r , ϕ , f ) exp ( i m ϕ ) d ϕ = β 0 0 ν [ β ν ( + k l + m ) * Ψ ν ; k , l ( + k l + m ) * ( r , f ) + ( 1 ϵ ν , k + l m ) β ν ( k + l m ) Ψ ν ; l , k ( k + l m ) ( r , f ) ] ,
Ψ a n m ( r , f ) Ψ m ( r , f ) .
( Ψ , Φ ) = 0 R F + F Ψ ( r , f ) Φ * ( r , f ) r d r d f .
I S = 1 π 0 2 π 0 1 A ( ρ , θ ) ρ d ρ d θ 2 1 π 0 2 π 0 1 A ( ρ , θ ) 2 ρ d ρ d θ .
A ( ρ , θ ) = 1 ( 1 s 0 2 ρ 2 ) 1 4 [ 1 + ( 1 s 0 2 ρ 2 ) 1 2 2 ] ,
I S = ( 8 75 s 0 2 ) [ 8 5 ( 1 s 0 2 ) 3 4 3 ( 1 s 0 2 ) 5 4 ] 2 4 + 3 s 0 2 ( 1 s 0 2 ) 1 2 { 4 s 0 2 } .
Ψ a n m ( r , f ) = β 0 0 ν { β ν m * [ Ψ ν ; 0 , 0 m * + s 0 2 ( Ψ ν ; 1 , 1 m * + Ψ ν ; 1 , 1 m * ) + s 0 4 2 ( Ψ ν ; 2 , 2 m * + Ψ ν ; 2 , 2 m * ) ] + β ν m ( 1 ϵ ν , m ) [ Ψ ν ; 0 , 0 m + s 0 2 ( Ψ ν ; 1 , 1 m + Ψ ν ; 1 , 1 m ) + s 0 4 2 ( Ψ ν ; 2 , 2 m + Ψ ν ; 2 , 2 m ) ] } ,
g Re m = β 0 0 2 ν ( Re [ β ν ( k l m ) * Ψ ν ; k , l ( k l m ) * ] + Re [ β ν ( k l + m ) * Ψ ν ; k , l ( k l + m ) * ] + ( 1 ϵ ν , k + l + m ) Re [ β ν ( k + l + m ) Ψ ν ; l , k ( k + l + m ) ] + ( 1 ϵ ν , k + l m ) Re [ β ν ( k + l m ) Ψ ν ; l , k ( k + l m ) ] i { Im [ β ν ( k l m ) * Ψ ν ; k , l ( k l m ) * ] Im [ β ν ( k l + m ) * Ψ ν ; k , l ( k l + m ) * ] + ( 1 ϵ ν , k + l + m ) Im [ β ν ( k + l + m ) Ψ ν ; l , k ( k + l + m ) ] ( 1 ϵ ν , k + l m ) Im [ β ν ( k + l m ) Ψ ν ; l , k ( k + l m ) ] } ) ,
g Im m = β 0 0 2 ν ( Im [ β ν ( k l m ) * Ψ ν ; k , l ( k l m ) * ] + Im [ β ν ( k l + m ) * Ψ ν ; k , l ( k l + m ) * ] + ( 1 ϵ ν , k + l + m ) Im [ β ν ( k + l + m ) Ψ ν ; l , k ( k + l + m ) ] + ( 1 ϵ ν , k + l m ) Im [ β ν ( k + l m ) Ψ ν ; l , k ( k + l m ) ] + i { Re [ β ν ( k l m ) * Ψ ν ; k , l ( k l m ) * ] Re [ β ν ( k l + m ) * Ψ ν ; k , l ( k l + m ) * ] + ( 1 ϵ ν , k + l + m ) Re [ β ν ( k + l + m ) Ψ ν ; l , k ( k + l + m ) ] ( 1 ϵ ν , k + l m ) Re [ β ν ( k + l m ) Ψ ν ; l , k ( k + l m ) ] } ) ,

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