Abstract

The use of the asymptotic treatment for the wedge diffraction problem established long ago by Pauli [Phys. Rev. 54, 924 (1938)] is here revisited and proposed in the character of a powerful computational tool for accurately retrieving the total electromagnetic field even in the near zone. After proving its factorial divergent character, the Pauli series is summed through the Weniger transformation, a nonlinear resummation scheme particularly efficient in the case of factorial divergence. Numerical results are carried out to show the accuracy and effectiveness of the proposed approach.

© 2008 Optical Society of America

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  1. A. Sommerfeld, Lectures on Theoretical Physics. Optics (Academic, 1964).
  2. Y. Umul, "Modified theory of physical optics approach to wedge diffraction problems," Opt. Express 13, 216-224 (2005).
    [CrossRef] [PubMed]
  3. Y. Umul, "Modified theory of the physical-optics approach to the impedance wedge problem," Opt. Lett. 31, 401-403 (2006).
    [CrossRef] [PubMed]
  4. M. Katsav and E. Heyman, "Phase space Gaussian beam summation analysis of half plane diffraction," IEEE Trans. Antennas Propag. 55, 1535-1545 (2007).
    [CrossRef]
  5. V. Nesterenko, G. Lambiase, and G. Scarpetta, "Casimir effect for a perfectly conducting wedge in terms of local zeta function," Ann. Phys. (N.Y.) 298, 403-420 (2002).
    [CrossRef]
  6. N. Pavloff and C. Schimt, "Diffractive orbits in quantum billiards," Phys. Rev. Lett. 75, 61-64 (1995).
    [CrossRef] [PubMed]
  7. H. Bruusy and N. Whelan, "Edge diffraction, trace formulae and the cardioid billiard," Nonlinearity 9, 1023-1047 (1996).
    [CrossRef]
  8. J. H. Hannay and A. Thain, "Exact scattering theory for any straight reflectors in two dimensions," J. Phys. A 36, 4063-4080 (2003).
    [CrossRef]
  9. W. Pauli, "On asymptotic series for functions in the theory of diffraction of light," Phys. Rev. 54, 924-931 (1938).
    [CrossRef]
  10. P. C. Clemmow, "Some extensions to the method of integration by steepest descents," Q. J. Mech. Appl. Math. 3, 241-256 (1950).
    [CrossRef]
  11. M. Alvarez, "Calculation of the Aharonov-Bohm wave function," Phys. Rev. A 54, 1128-1135 (1996).
    [CrossRef] [PubMed]
  12. M. V. Vesnik, "Analytical solution for electromagnetic diffraction on 2-D perfectly conducting scatterers of arbitrary shape," IEEE Trans. Antennas Propag. 49, 1638-1644 (2001).
    [CrossRef]
  13. Darboux, "Mémoire sur l'approximation de fonctions de très grande nombres et sur une classe étendue de développments en séries," J. Math. Pures Appl. 4, 5-56 (1878).
  14. R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, 1973).
  15. G. H. Hardy, Divergent Series (AMS, 1991).
  16. E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10, 189-371 (1989).
    [CrossRef]
  17. J. Cizek, J. Zamastil, and L. Skála, "New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field," J. Math. Phys. 44, 962-968 (2003).
    [CrossRef]
  18. E. J. Weniger, "Mathematical properties of a new Levin-type sequence transformation introduced by Cizek, Zamastil, and Skala. I. Algebraic theory," J. Math. Phys. 45, 1209-1246 (2004).
    [CrossRef]
  19. E. J. Weniger, "Asymptotic approximations to truncation errors of series representations for special functions," arXiv.org e-Print archive, math.CA/0511074v1, 11 March 2005, http://arxiv.org/abs/math/0511074v1.
  20. For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.
    [CrossRef]
  21. It should be noted that in the original paper, Pauli used the opposite convention; i. e., the temporal factor had the form exp(+iωt). For this reason, the subsequent formulas are the complex conjugate of those written in .
  22. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  23. F. Gori, "Diffraction from a half-plane. A new derivation of the Sommerfeld solution," Opt. Commun. 48, 67-70 (1983).
    [CrossRef]
  24. We use, as far as possible, a notation similar to the original one adopted in .
  25. If, on the contrary, the illumination produces no geometrical shadow, to obtain a regular representation also at the second reflection, it is sufficient to change φ into φ−2πn in all the following equations.
  26. M.Abramowitz and I.Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).
  27. M. V. Berry, "Universal oscillations of high derivatives," Proc. R. Soc. London, Ser. A 461, 1735-1751 (2005).
    [CrossRef]
  28. When z is a real negative, the integral in Eq. must be intended in the Cauchy principal value sense. This is related to the fact that the line arg z=π represents a branch cut for the exponential integral function .
  29. R. Borghi and M. Santarsiero, "Summing Lax series for nonparaxial propagation," Opt. Lett. 28, 774-776 (2003).
    [CrossRef] [PubMed]
  30. R. Borghi, "Evaluation of diffraction catastrophes by using Weniger transformation," Opt. Lett. 32, 226-228 (2007).
    [CrossRef] [PubMed]
  31. R. Borghi, "Optical asymptotics via Weniger transformation," arXiv.org e-Print archieve, physics/0706.3573, 25 June 2007, http://arxiv.org/abs/0706.3573.
  32. M. A. Alonso and R. Borghi, "Complete far-field asymptotic series for free fields," Opt. Lett. 31, 3028-3030 (2006).
    [CrossRef] [PubMed]

2007

M. Katsav and E. Heyman, "Phase space Gaussian beam summation analysis of half plane diffraction," IEEE Trans. Antennas Propag. 55, 1535-1545 (2007).
[CrossRef]

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.
[CrossRef]

R. Borghi, "Evaluation of diffraction catastrophes by using Weniger transformation," Opt. Lett. 32, 226-228 (2007).
[CrossRef] [PubMed]

2006

2005

Y. Umul, "Modified theory of physical optics approach to wedge diffraction problems," Opt. Express 13, 216-224 (2005).
[CrossRef] [PubMed]

M. V. Berry, "Universal oscillations of high derivatives," Proc. R. Soc. London, Ser. A 461, 1735-1751 (2005).
[CrossRef]

2004

E. J. Weniger, "Mathematical properties of a new Levin-type sequence transformation introduced by Cizek, Zamastil, and Skala. I. Algebraic theory," J. Math. Phys. 45, 1209-1246 (2004).
[CrossRef]

2003

R. Borghi and M. Santarsiero, "Summing Lax series for nonparaxial propagation," Opt. Lett. 28, 774-776 (2003).
[CrossRef] [PubMed]

J. H. Hannay and A. Thain, "Exact scattering theory for any straight reflectors in two dimensions," J. Phys. A 36, 4063-4080 (2003).
[CrossRef]

J. Cizek, J. Zamastil, and L. Skála, "New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field," J. Math. Phys. 44, 962-968 (2003).
[CrossRef]

2002

V. Nesterenko, G. Lambiase, and G. Scarpetta, "Casimir effect for a perfectly conducting wedge in terms of local zeta function," Ann. Phys. (N.Y.) 298, 403-420 (2002).
[CrossRef]

2001

M. V. Vesnik, "Analytical solution for electromagnetic diffraction on 2-D perfectly conducting scatterers of arbitrary shape," IEEE Trans. Antennas Propag. 49, 1638-1644 (2001).
[CrossRef]

1999

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

1996

M. Alvarez, "Calculation of the Aharonov-Bohm wave function," Phys. Rev. A 54, 1128-1135 (1996).
[CrossRef] [PubMed]

H. Bruusy and N. Whelan, "Edge diffraction, trace formulae and the cardioid billiard," Nonlinearity 9, 1023-1047 (1996).
[CrossRef]

1995

N. Pavloff and C. Schimt, "Diffractive orbits in quantum billiards," Phys. Rev. Lett. 75, 61-64 (1995).
[CrossRef] [PubMed]

1991

G. H. Hardy, Divergent Series (AMS, 1991).

1989

E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10, 189-371 (1989).
[CrossRef]

1983

F. Gori, "Diffraction from a half-plane. A new derivation of the Sommerfeld solution," Opt. Commun. 48, 67-70 (1983).
[CrossRef]

1973

R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, 1973).

1972

M.Abramowitz and I.Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).

1964

A. Sommerfeld, Lectures on Theoretical Physics. Optics (Academic, 1964).

1950

P. C. Clemmow, "Some extensions to the method of integration by steepest descents," Q. J. Mech. Appl. Math. 3, 241-256 (1950).
[CrossRef]

1938

W. Pauli, "On asymptotic series for functions in the theory of diffraction of light," Phys. Rev. 54, 924-931 (1938).
[CrossRef]

1878

Darboux, "Mémoire sur l'approximation de fonctions de très grande nombres et sur une classe étendue de développments en séries," J. Math. Pures Appl. 4, 5-56 (1878).

Alonso, M. A.

Alvarez, M.

M. Alvarez, "Calculation of the Aharonov-Bohm wave function," Phys. Rev. A 54, 1128-1135 (1996).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry, "Universal oscillations of high derivatives," Proc. R. Soc. London, Ser. A 461, 1735-1751 (2005).
[CrossRef]

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Bruusy, H.

H. Bruusy and N. Whelan, "Edge diffraction, trace formulae and the cardioid billiard," Nonlinearity 9, 1023-1047 (1996).
[CrossRef]

Caliceti, E.

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.
[CrossRef]

Cizek, J.

J. Cizek, J. Zamastil, and L. Skála, "New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field," J. Math. Phys. 44, 962-968 (2003).
[CrossRef]

Clemmow, P. C.

P. C. Clemmow, "Some extensions to the method of integration by steepest descents," Q. J. Mech. Appl. Math. 3, 241-256 (1950).
[CrossRef]

Dingle, R. B.

R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, 1973).

Gori, F.

F. Gori, "Diffraction from a half-plane. A new derivation of the Sommerfeld solution," Opt. Commun. 48, 67-70 (1983).
[CrossRef]

Hannay, J. H.

J. H. Hannay and A. Thain, "Exact scattering theory for any straight reflectors in two dimensions," J. Phys. A 36, 4063-4080 (2003).
[CrossRef]

Hardy, G. H.

G. H. Hardy, Divergent Series (AMS, 1991).

Heyman, E.

M. Katsav and E. Heyman, "Phase space Gaussian beam summation analysis of half plane diffraction," IEEE Trans. Antennas Propag. 55, 1535-1545 (2007).
[CrossRef]

Jentschura, U. D.

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.
[CrossRef]

Katsav, M.

M. Katsav and E. Heyman, "Phase space Gaussian beam summation analysis of half plane diffraction," IEEE Trans. Antennas Propag. 55, 1535-1545 (2007).
[CrossRef]

Lambiase, G.

V. Nesterenko, G. Lambiase, and G. Scarpetta, "Casimir effect for a perfectly conducting wedge in terms of local zeta function," Ann. Phys. (N.Y.) 298, 403-420 (2002).
[CrossRef]

Meyer-Hermann, M.

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.
[CrossRef]

Nesterenko, V.

V. Nesterenko, G. Lambiase, and G. Scarpetta, "Casimir effect for a perfectly conducting wedge in terms of local zeta function," Ann. Phys. (N.Y.) 298, 403-420 (2002).
[CrossRef]

Pauli, W.

W. Pauli, "On asymptotic series for functions in the theory of diffraction of light," Phys. Rev. 54, 924-931 (1938).
[CrossRef]

Pavloff, N.

N. Pavloff and C. Schimt, "Diffractive orbits in quantum billiards," Phys. Rev. Lett. 75, 61-64 (1995).
[CrossRef] [PubMed]

Ribeca, P.

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.
[CrossRef]

Santarsiero, M.

Scarpetta, G.

V. Nesterenko, G. Lambiase, and G. Scarpetta, "Casimir effect for a perfectly conducting wedge in terms of local zeta function," Ann. Phys. (N.Y.) 298, 403-420 (2002).
[CrossRef]

Schimt, C.

N. Pavloff and C. Schimt, "Diffractive orbits in quantum billiards," Phys. Rev. Lett. 75, 61-64 (1995).
[CrossRef] [PubMed]

Skála, L.

J. Cizek, J. Zamastil, and L. Skála, "New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field," J. Math. Phys. 44, 962-968 (2003).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, Lectures on Theoretical Physics. Optics (Academic, 1964).

Surzhykov, A.

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.
[CrossRef]

Thain, A.

J. H. Hannay and A. Thain, "Exact scattering theory for any straight reflectors in two dimensions," J. Phys. A 36, 4063-4080 (2003).
[CrossRef]

Umul, Y.

Vesnik, M. V.

M. V. Vesnik, "Analytical solution for electromagnetic diffraction on 2-D perfectly conducting scatterers of arbitrary shape," IEEE Trans. Antennas Propag. 49, 1638-1644 (2001).
[CrossRef]

Weniger, E. J.

E. J. Weniger, "Mathematical properties of a new Levin-type sequence transformation introduced by Cizek, Zamastil, and Skala. I. Algebraic theory," J. Math. Phys. 45, 1209-1246 (2004).
[CrossRef]

E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10, 189-371 (1989).
[CrossRef]

E. J. Weniger, "Asymptotic approximations to truncation errors of series representations for special functions," arXiv.org e-Print archive, math.CA/0511074v1, 11 March 2005, http://arxiv.org/abs/math/0511074v1.

Whelan, N.

H. Bruusy and N. Whelan, "Edge diffraction, trace formulae and the cardioid billiard," Nonlinearity 9, 1023-1047 (1996).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Zamastil, J.

J. Cizek, J. Zamastil, and L. Skála, "New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field," J. Math. Phys. 44, 962-968 (2003).
[CrossRef]

Ann. Phys. (N.Y.)

V. Nesterenko, G. Lambiase, and G. Scarpetta, "Casimir effect for a perfectly conducting wedge in terms of local zeta function," Ann. Phys. (N.Y.) 298, 403-420 (2002).
[CrossRef]

Comput. Phys. Rep.

E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series," Comput. Phys. Rep. 10, 189-371 (1989).
[CrossRef]

IEEE Trans. Antennas Propag.

M. V. Vesnik, "Analytical solution for electromagnetic diffraction on 2-D perfectly conducting scatterers of arbitrary shape," IEEE Trans. Antennas Propag. 49, 1638-1644 (2001).
[CrossRef]

M. Katsav and E. Heyman, "Phase space Gaussian beam summation analysis of half plane diffraction," IEEE Trans. Antennas Propag. 55, 1535-1545 (2007).
[CrossRef]

J. Math. Phys.

J. Cizek, J. Zamastil, and L. Skála, "New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field," J. Math. Phys. 44, 962-968 (2003).
[CrossRef]

E. J. Weniger, "Mathematical properties of a new Levin-type sequence transformation introduced by Cizek, Zamastil, and Skala. I. Algebraic theory," J. Math. Phys. 45, 1209-1246 (2004).
[CrossRef]

J. Math. Pures Appl.

Darboux, "Mémoire sur l'approximation de fonctions de très grande nombres et sur une classe étendue de développments en séries," J. Math. Pures Appl. 4, 5-56 (1878).

J. Phys. A

J. H. Hannay and A. Thain, "Exact scattering theory for any straight reflectors in two dimensions," J. Phys. A 36, 4063-4080 (2003).
[CrossRef]

Nonlinearity

H. Bruusy and N. Whelan, "Edge diffraction, trace formulae and the cardioid billiard," Nonlinearity 9, 1023-1047 (1996).
[CrossRef]

Opt. Commun.

F. Gori, "Diffraction from a half-plane. A new derivation of the Sommerfeld solution," Opt. Commun. 48, 67-70 (1983).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

For an updated review about methods for summing diverging series, see for instance E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, and U. D. Jentschura, "From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions," Phys. Rep. 446, 1-96 (2007), arXiv.org e-Print archieve, physics/0707.1596v1, 11 July 2007, http://arxiv.org/abs/0707.1596v1.
[CrossRef]

Phys. Rev.

W. Pauli, "On asymptotic series for functions in the theory of diffraction of light," Phys. Rev. 54, 924-931 (1938).
[CrossRef]

Phys. Rev. A

M. Alvarez, "Calculation of the Aharonov-Bohm wave function," Phys. Rev. A 54, 1128-1135 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett.

N. Pavloff and C. Schimt, "Diffractive orbits in quantum billiards," Phys. Rev. Lett. 75, 61-64 (1995).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A

M. V. Berry, "Universal oscillations of high derivatives," Proc. R. Soc. London, Ser. A 461, 1735-1751 (2005).
[CrossRef]

Q. J. Mech. Appl. Math.

P. C. Clemmow, "Some extensions to the method of integration by steepest descents," Q. J. Mech. Appl. Math. 3, 241-256 (1950).
[CrossRef]

Other

A. Sommerfeld, Lectures on Theoretical Physics. Optics (Academic, 1964).

R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, 1973).

G. H. Hardy, Divergent Series (AMS, 1991).

E. J. Weniger, "Asymptotic approximations to truncation errors of series representations for special functions," arXiv.org e-Print archive, math.CA/0511074v1, 11 March 2005, http://arxiv.org/abs/math/0511074v1.

When z is a real negative, the integral in Eq. must be intended in the Cauchy principal value sense. This is related to the fact that the line arg z=π represents a branch cut for the exponential integral function .

R. Borghi, "Optical asymptotics via Weniger transformation," arXiv.org e-Print archieve, physics/0706.3573, 25 June 2007, http://arxiv.org/abs/0706.3573.

It should be noted that in the original paper, Pauli used the opposite convention; i. e., the temporal factor had the form exp(+iωt). For this reason, the subsequent formulas are the complex conjugate of those written in .

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

We use, as far as possible, a notation similar to the original one adopted in .

If, on the contrary, the illumination produces no geometrical shadow, to obtain a regular representation also at the second reflection, it is sufficient to change φ into φ−2πn in all the following equations.

M.Abramowitz and I.Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).

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Figures (5)

Fig. 1
Fig. 1

Geometry for the wedge problem.

Fig. 2
Fig. 2

Behavior of the relative error for the total electric field produced by the scattering of an E-polarized plane wave, impinging at an angle φ 0 = 3 π 4 on a wedge having θ = π 2 ( n = 3 2 ) , evaluated at a point placed along the direction φ = π , at a distance of λ 8 from the edge of the wedge. In (a) the error obtained by using the WT on the sequence of the partial sums is plotted, whereas in (b) the error obtained by using only the partial sums is also shown.

Fig. 3
Fig. 3

Behavior of the relative error for the same case of Fig. 2, evaluated as a function of the distance from the edge of the wedge, ranging from λ 8 to 10 λ , along the line φ = π . Different values of the WT order have been used to evaluated the total field. In particular, the solid curve corresponds to m = 10 , and the dashed curve to m = 4 . Furthermore, it is also reported, as a dotted curve, the error obtained by merely summing the first two terms of the Pauli series without the application of any resummation scheme.

Fig. 4
Fig. 4

Behavior of the relative error for the total electric field produced by the scattering of an E-polarized plane wave, impinging at an angle φ 0 = π 3 on a wedge having θ = π 6 ( n = 11 6 ) , evaluated at points located on a circle centered at the edge of the wedge having radius 6 λ . The total field is evaluated through δ m in Eq. (26) with m = 4 (dashed curve) and m = 10 (solid curve). The values of the relative error, obtained directly through the use of the first two terms of the Pauli series, are also shown as a dotted curve.

Fig. 5
Fig. 5

Two-dimensional maps as functions of x λ and y λ of the modulus (first column) and the phase (second column) of the total electric field for a wedge having θ = π 6 ( n = 11 6 ) , illuminated by an E-polarized plane wave impinging at φ 0 = π 4 (first row), φ 0 = 11 π 12 (second row), and φ 0 = 5 π 4 (third row). The values of the diffracted field V d are obtained by summing the Pauli series through δ 10 . Thick curves in the phase maps represent the loci at which the phase undergoes the wrapping action.

Equations (34)

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

U 0 ( r , φ ) = exp [ i r cos ( φ φ 0 ) ] ,
U ( r , φ ) = V ( r , φ φ 0 ) ± V ( r , φ + φ 0 ) ,
π n = 2 π θ ,
V ( r , φ ) = V g ( r , φ ) + V d ( r , φ ) ,
V g ( r , φ ) = { exp [ i r cos ( φ + 2 π n N ) ] π < φ + 2 π n N < π 0 otherwise } ,
π + φ 2 π n < N < π φ 2 π n ,
V d ( r , φ ) = 1 2 π i 1 n sin π n i + ϵ + i ϵ exp ( i r cos η ) cos π n cos η + φ n d η ,
V d ( r , φ ) exp [ i ( r + π 4 ) ] 2 π r 1 n sin π n cos π n cos φ n ,
cos φ n = cos π n .
φ = ± π + 2 π n N ,
s 2 = i ( 1 cos η ) ,
V d ( r , φ ) = 2 exp [ i ( r π 4 ) ] 2 π 1 n sin π n + exp ( r s 2 ) f ( s , φ ) d s s 2 i a ,
f ( s , φ ) = cos η + cos φ cos π n cos η + φ n 1 cos η 2 ,
f ( s , φ ) = m = 0 exp ( i m π 4 ) A m ( φ ) s m ,
+ exp ( r s 2 ) s 2 m s 2 i a d s = exp ( i a r ) r 1 2 m Γ ( m + 1 2 ) E m + 1 2 ( i a r ) ,
V d ( r , φ ) = 1 2 π i m = 0 exp ( i r ) ( i r ) m + 1 2 P m ( r , φ ) ,
P m ( r , φ ) = 2 n sin π n ( m 1 2 ) ! A 2 m ( φ ) i r exp ( i a r ) E m + 1 2 ( i a r ) .
A 2 m ( φ ) = i m ( 2 m ) ! [ 2 m f ( s , φ ) s 2 m ] s = 0 .
exp ( z ) E m + 1 2 ( z ) 1 m .
A 2 m ( φ ) = 2 m ( 2 m ) ! [ 2 m F ( ξ , φ ) ξ 2 m ] ξ = 0 ,
F ( ξ , φ ) = 1 + cos φ 2 ξ 2 cos π n cos 2 arcsin ξ + φ n 1 1 ξ 2 ,
A 2 m ( φ ) 2 π cos π n cos φ 1 cos 2 π n cos π + φ n ( 2 m 1 2 ) ! 2 m ( 2 m ) ! ,
E ( z ) = m = 0 ( 1 ) m z m m ! .
E ( z ) = 0 d u exp ( u ) m = 0 ( z u ) m ,
E ( z ) = 0 exp ( u ) 1 + z u d u = exp ( 1 z ) z E 1 ( 1 z ) ,
δ k = j = 0 k ( 1 ) j ( k j ) ( 1 + j ) k 1 s j a j + 1 j = 0 k ( 1 ) j ( k j ) ( 1 + j ) k 1 1 a j + 1 ,
f ( z ) = m = 0 a m z m ,
a m = f ( m ) ( 0 ) m ! .
f ( z ) = ( z s z ) α g ( z ) ,
g ( z ) = m = 0 g s ( m ) m ! ( z z s ) m ,
( z s z ) α = 1 ( α 1 ) ! z s α j = 0 ( α + j 1 ) ! j ! ( z z s ) j .
a m = ( α + m 1 ) ! m ! ( α 1 ) ! 1 z s α + m [ g s ( 0 ) α 1 m + α 1 z s g s ( 1 ) + ( α 1 ) ( α 2 ) 2 ! ( m + α 1 ) ( m + α 2 ) z s 2 g s ( 2 ) ] .
[ d m f d z m ] z = 0 g s ( 0 ) z s α + m ( α + m 1 ) ! ( α 1 ) ! ,
g s ( 0 ) = lim z z s ( z s z ) α f ( z ) .

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