E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39, 204–210 (1981), and references therein.

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976).

N. Chako, "Asymptotic expansions of double and multiple integrals occurring in diffraction theory," J. Inst. Math. Appl. 1, 372–422 (1965), Sec. 3.

D. S. Jones and M. Kline, "Asymptotic expansion of multiple integrals and the method of stationary phase," J. Math. Phys. 37, 1–28 (1958).

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpuktes oder iener Brennlinie," Ann. Phys. 30, 755–776 (1909).

M. Born and W. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1980), Sec. 8.8.

H. Bremermann, Distributions, Complex Variables and Fourier Transforms (Addison-Wesley, Reading, Mass., 1965), Sec. 3.3–3.4.

N. Chako, "Asymptotic expansions of double and multiple integrals occurring in diffraction theory," J. Inst. Math. Appl. 1, 372–422 (1965), Sec. 3.

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpuktes oder iener Brennlinie," Ann. Phys. 30, 755–776 (1909).

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), Sec. 2.8–2.9.

D. S. Jones and M. Kline, "Asymptotic expansion of multiple integrals and the method of stationary phase," J. Math. Phys. 37, 1–28 (1958).

D. S. Jones and M. Kline, "Asymptotic expansion of multiple integrals and the method of stationary phase," J. Math. Phys. 37, 1–28 (1958).

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976).

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39, 204–210 (1981), and references therein.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Sec 46.

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976).

A. Sommerfeld, Optics, Vol. IV of Lectures on Theoretical Physics (Academic, New York, 1964), Sec. 45.

A. Sommerfeld, Partial Differential Equations in Physics, Vol. VI of Lectures on Theoretical Physics (Academic, New York, 1964), Sec. 28.

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976).

G. Tyras, Radiation and Propagation of Electromagnetic Waves (Academic, New York, 1969). Our analysis here parallels the analysis of a similar integral presented in Sec. 5.1.1 of this reference.

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpuktes oder iener Brennlinie," Ann. Phys. 30, 755–776 (1909).

N. Chako, "Asymptotic expansions of double and multiple integrals occurring in diffraction theory," J. Inst. Math. Appl. 1, 372–422 (1965), Sec. 3.

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976).

D. S. Jones and M. Kline, "Asymptotic expansion of multiple integrals and the method of stationary phase," J. Math. Phys. 37, 1–28 (1958).

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39, 204–210 (1981), and references therein.

M. Born and W. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1980), Sec. 8.8.

G. Tyras, Radiation and Propagation of Electromagnetic Waves (Academic, New York, 1969). Our analysis here parallels the analysis of a similar integral presented in Sec. 5.1.1 of this reference.

Ref. 15, Eq. (5.35b).

A. Sommerfeld, Partial Differential Equations in Physics, Vol. VI of Lectures on Theoretical Physics (Academic, New York, 1964), Sec. 28.

A. Sommerfeld, Optics, Vol. IV of Lectures on Theoretical Physics (Academic, New York, 1964), Sec. 45.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Sec 46.

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), Sec. 2.8–2.9.

H. Bremermann, Distributions, Complex Variables and Fourier Transforms (Addison-Wesley, Reading, Mass., 1965), Sec. 3.3–3.4.

This is done for the critical points on the corners in Ref. 8. The same arguments can be applied to the other critical points.