Abstract

A scale transformation that converts an ellipse into a circle has been suggested in the literature as a method for eliminating the need to evaluate the conventional Mathieu function solution for scattering by an elliptic cylinder. This suggestion is tested by examining the wave equation in the scaled coordinate system and by evaluating the scattering from a thin ellipse for conditions where it is expected that an approximate solution can be obtained using the scalar theory single-slit approximation. It is found that, for a plane electromagnetic wave normally incident on a thin perfectly conducting ellipse, the position of the first minimum in the diffraction pattern, relative to the central axis, differs by approximately a factor of 7 between the single-slit and the scaled theory approach to the problem. The examination of the scaled wave equation and the scattering calculation suggests that, because the scale transformation generates an anisotropic medium, the use of a uniform medium solution in the scaled coordinate system is not appropriate.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).
  2. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195(1955).
    [CrossRef]
  3. B. Sieger, “Die beugung einer ebenen elektrischen welle an einem schirm von elliptischem querschnitt,” Ann. Phys. 332, 626–664 (1908).
    [CrossRef]
  4. P. M. Morse and P. L. Rubenstein, “The diffraction of waves by ribbons and by slits,” Phys. Rev. 54, 895–898(1938).
    [CrossRef]
  5. C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65–71 (1963).
    [CrossRef]
  6. C. Yeh, “Scattering of obliquely incident light waves by elliptical fibers,” J. Opt. Soc. Am. 54, 1227–1281 (1964).
    [CrossRef]
  7. N. W. McLachlan, Theory and Application of Mathieu Functions, Corrected ed. (Oxford University Press, 1951), pp. 363–365.
  8. G. J. Burke and H. Pao, “Evaluation of the Mathieu function series for diffraction by a slot,” Lawrence Livermore National Laboratory report UCRL-TR-213076 (2005).
  9. Maple 11 does not evaluate the Modified Mathieu function correctly; this has not been corrected in Maple 14.
  10. A. L. Van Buren and J. E. Boisvert, “Accurate calculation of the modified Mathieu functions of integer order,” Q. Appl. Math. 65, 1–21 (2007).
  11. J.-T. Zhang and Y.-L. Li, “A new research approach of electromagnetic theory and its applications,” Indian J. Radio Space Phys. 35, 249–252 (2006).
  12. Y. L. Li, J. Y. Huang, M. J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
    [CrossRef]
  13. Y. L. Li, M. J. Wang, and G. F. Tang, “The scattering from an elliptic cylinder irradiated by an electromagnetic wave with arbitrary direction and polarization,” Prog. Electromagn. Res. Lett. 5, 137–149 (2008).
    [CrossRef]
  14. A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
    [CrossRef]
  15. S. G. Johnson, “Coordinate transformation and invariance in electromagnetism,” http://www-math.mit.edu/~stevenj/18.369/coordinate-transform.pdf.
  16. C. Kottke, A. Farjadpour, and S. G. Johnson, “Perturbation theory for anisotropic dielectric interfaces and application to subpixel smoothing of discretized numerical methods,” Phys. Rev. E 77, 036611 (2008).
    [CrossRef]
  17. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 3rd ed., revised (Peregrinus, 1986), pp. 74–76.
  18. J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986), pp. 489–491.
  19. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2003), pp. 421–637.
  20. R. G. Greenler, J. W. Hable, and P. O. Slane, “Diffraction around a fine wire: how good is the single slit approximation?” Am. J. Phys. 58, 330–331 (1990).
    [CrossRef]

2008 (3)

Y. L. Li, J. Y. Huang, M. J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
[CrossRef]

Y. L. Li, M. J. Wang, and G. F. Tang, “The scattering from an elliptic cylinder irradiated by an electromagnetic wave with arbitrary direction and polarization,” Prog. Electromagn. Res. Lett. 5, 137–149 (2008).
[CrossRef]

C. Kottke, A. Farjadpour, and S. G. Johnson, “Perturbation theory for anisotropic dielectric interfaces and application to subpixel smoothing of discretized numerical methods,” Phys. Rev. E 77, 036611 (2008).
[CrossRef]

2007 (1)

A. L. Van Buren and J. E. Boisvert, “Accurate calculation of the modified Mathieu functions of integer order,” Q. Appl. Math. 65, 1–21 (2007).

2006 (1)

J.-T. Zhang and Y.-L. Li, “A new research approach of electromagnetic theory and its applications,” Indian J. Radio Space Phys. 35, 249–252 (2006).

1996 (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[CrossRef]

1990 (1)

R. G. Greenler, J. W. Hable, and P. O. Slane, “Diffraction around a fine wire: how good is the single slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

1964 (1)

1963 (1)

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65–71 (1963).
[CrossRef]

1955 (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195(1955).
[CrossRef]

1938 (1)

P. M. Morse and P. L. Rubenstein, “The diffraction of waves by ribbons and by slits,” Phys. Rev. 54, 895–898(1938).
[CrossRef]

1908 (1)

B. Sieger, “Die beugung einer ebenen elektrischen welle an einem schirm von elliptischem querschnitt,” Ann. Phys. 332, 626–664 (1908).
[CrossRef]

1881 (1)

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

Boisvert, J. E.

A. L. Van Buren and J. E. Boisvert, “Accurate calculation of the modified Mathieu functions of integer order,” Q. Appl. Math. 65, 1–21 (2007).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2003), pp. 421–637.

Burke, G. J.

G. J. Burke and H. Pao, “Evaluation of the Mathieu function series for diffraction by a slot,” Lawrence Livermore National Laboratory report UCRL-TR-213076 (2005).

Farjadpour, A.

C. Kottke, A. Farjadpour, and S. G. Johnson, “Perturbation theory for anisotropic dielectric interfaces and application to subpixel smoothing of discretized numerical methods,” Phys. Rev. E 77, 036611 (2008).
[CrossRef]

Greenler, R. G.

R. G. Greenler, J. W. Hable, and P. O. Slane, “Diffraction around a fine wire: how good is the single slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

Hable, J. W.

R. G. Greenler, J. W. Hable, and P. O. Slane, “Diffraction around a fine wire: how good is the single slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

Huang, J. Y.

Y. L. Li, J. Y. Huang, M. J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
[CrossRef]

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 3rd ed., revised (Peregrinus, 1986), pp. 74–76.

Johnson, S. G.

C. Kottke, A. Farjadpour, and S. G. Johnson, “Perturbation theory for anisotropic dielectric interfaces and application to subpixel smoothing of discretized numerical methods,” Phys. Rev. E 77, 036611 (2008).
[CrossRef]

S. G. Johnson, “Coordinate transformation and invariance in electromagnetism,” http://www-math.mit.edu/~stevenj/18.369/coordinate-transform.pdf.

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986), pp. 489–491.

Kottke, C.

C. Kottke, A. Farjadpour, and S. G. Johnson, “Perturbation theory for anisotropic dielectric interfaces and application to subpixel smoothing of discretized numerical methods,” Phys. Rev. E 77, 036611 (2008).
[CrossRef]

Li, Y. L.

Y. L. Li, J. Y. Huang, M. J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
[CrossRef]

Y. L. Li, M. J. Wang, and G. F. Tang, “The scattering from an elliptic cylinder irradiated by an electromagnetic wave with arbitrary direction and polarization,” Prog. Electromagn. Res. Lett. 5, 137–149 (2008).
[CrossRef]

Li, Y.-L.

J.-T. Zhang and Y.-L. Li, “A new research approach of electromagnetic theory and its applications,” Indian J. Radio Space Phys. 35, 249–252 (2006).

McLachlan, N. W.

N. W. McLachlan, Theory and Application of Mathieu Functions, Corrected ed. (Oxford University Press, 1951), pp. 363–365.

Morse, P. M.

P. M. Morse and P. L. Rubenstein, “The diffraction of waves by ribbons and by slits,” Phys. Rev. 54, 895–898(1938).
[CrossRef]

Pao, H.

G. J. Burke and H. Pao, “Evaluation of the Mathieu function series for diffraction by a slot,” Lawrence Livermore National Laboratory report UCRL-TR-213076 (2005).

Pendry, J. B.

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

Rubenstein, P. L.

P. M. Morse and P. L. Rubenstein, “The diffraction of waves by ribbons and by slits,” Phys. Rev. 54, 895–898(1938).
[CrossRef]

Sieger, B.

B. Sieger, “Die beugung einer ebenen elektrischen welle an einem schirm von elliptischem querschnitt,” Ann. Phys. 332, 626–664 (1908).
[CrossRef]

Slane, P. O.

R. G. Greenler, J. W. Hable, and P. O. Slane, “Diffraction around a fine wire: how good is the single slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

Tang, G. F.

Y. L. Li, M. J. Wang, and G. F. Tang, “The scattering from an elliptic cylinder irradiated by an electromagnetic wave with arbitrary direction and polarization,” Prog. Electromagn. Res. Lett. 5, 137–149 (2008).
[CrossRef]

Van Buren, A. L.

A. L. Van Buren and J. E. Boisvert, “Accurate calculation of the modified Mathieu functions of integer order,” Q. Appl. Math. 65, 1–21 (2007).

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195(1955).
[CrossRef]

Wang, M. J.

Y. L. Li, M. J. Wang, and G. F. Tang, “The scattering from an elliptic cylinder irradiated by an electromagnetic wave with arbitrary direction and polarization,” Prog. Electromagn. Res. Lett. 5, 137–149 (2008).
[CrossRef]

Y. L. Li, J. Y. Huang, M. J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
[CrossRef]

Ward, A. J.

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2003), pp. 421–637.

Yeh, C.

C. Yeh, “Scattering of obliquely incident light waves by elliptical fibers,” J. Opt. Soc. Am. 54, 1227–1281 (1964).
[CrossRef]

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65–71 (1963).
[CrossRef]

Zhang, J. T.

Y. L. Li, J. Y. Huang, M. J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
[CrossRef]

Zhang, J.-T.

J.-T. Zhang and Y.-L. Li, “A new research approach of electromagnetic theory and its applications,” Indian J. Radio Space Phys. 35, 249–252 (2006).

Am. J. Phys. (1)

R. G. Greenler, J. W. Hable, and P. O. Slane, “Diffraction around a fine wire: how good is the single slit approximation?” Am. J. Phys. 58, 330–331 (1990).
[CrossRef]

Ann. Phys. (1)

B. Sieger, “Die beugung einer ebenen elektrischen welle an einem schirm von elliptischem querschnitt,” Ann. Phys. 332, 626–664 (1908).
[CrossRef]

Can. J. Phys. (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195(1955).
[CrossRef]

Indian J. Radio Space Phys. (1)

J.-T. Zhang and Y.-L. Li, “A new research approach of electromagnetic theory and its applications,” Indian J. Radio Space Phys. 35, 249–252 (2006).

J. Math. Phys. (1)

C. Yeh, “The diffraction of waves by a penetrable ribbon,” J. Math. Phys. 4, 65–71 (1963).
[CrossRef]

J. Mod. Opt. (1)

A. J. Ward and J. B. Pendry, “Refraction and geometry in Maxwell’s equations,” J. Mod. Opt. 43, 773–793 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

Philos. Mag. (1)

Lord Rayleigh, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

Phys. Rev. (1)

P. M. Morse and P. L. Rubenstein, “The diffraction of waves by ribbons and by slits,” Phys. Rev. 54, 895–898(1938).
[CrossRef]

Phys. Rev. E (1)

C. Kottke, A. Farjadpour, and S. G. Johnson, “Perturbation theory for anisotropic dielectric interfaces and application to subpixel smoothing of discretized numerical methods,” Phys. Rev. E 77, 036611 (2008).
[CrossRef]

Prog. Electromagn. Res. Lett. (2)

Y. L. Li, J. Y. Huang, M. J. Wang, and J. T. Zhang, “Scattering field for the ellipsoidal targets irradiated by an electromagnetic wave with arbitrary polarizing and propagating direction,” Prog. Electromagn. Res. Lett. 1, 221–235 (2008).
[CrossRef]

Y. L. Li, M. J. Wang, and G. F. Tang, “The scattering from an elliptic cylinder irradiated by an electromagnetic wave with arbitrary direction and polarization,” Prog. Electromagn. Res. Lett. 5, 137–149 (2008).
[CrossRef]

Q. Appl. Math. (1)

A. L. Van Buren and J. E. Boisvert, “Accurate calculation of the modified Mathieu functions of integer order,” Q. Appl. Math. 65, 1–21 (2007).

Other (7)

S. G. Johnson, “Coordinate transformation and invariance in electromagnetism,” http://www-math.mit.edu/~stevenj/18.369/coordinate-transform.pdf.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 3rd ed., revised (Peregrinus, 1986), pp. 74–76.

J. A. Kong, Electromagnetic Wave Theory (Wiley, 1986), pp. 489–491.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2003), pp. 421–637.

N. W. McLachlan, Theory and Application of Mathieu Functions, Corrected ed. (Oxford University Press, 1951), pp. 363–365.

G. J. Burke and H. Pao, “Evaluation of the Mathieu function series for diffraction by a slot,” Lawrence Livermore National Laboratory report UCRL-TR-213076 (2005).

Maple 11 does not evaluate the Modified Mathieu function correctly; this has not been corrected in Maple 14.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

(a) Elliptic cylinder with major diameter 2 a and minor diameter 2 b in the Σ system. The wave vector k of the incident plane wave is inclined an angle ϕ o with respect to the x axis. (b) Scaled, or Σ , coordinate system with the ellipse of (a) transformed into a circle of radius R . The wave vector k of the incident plane wave is inclined at an angle ϕ o with respect to the x axis. The Σ system is rotated an angle ϕ o with respect to the x axis. In this system k is parallel to the x axis.

Fig. 2
Fig. 2

Solid curves show the amplitude of the scattered field from a thin elliptic cylinder obtained by evaluating the scaled theory formula, Eq. (35), as a function of the angle ϕ in the Σ system for two different values of 2 b / λ with ρ = 5.2 m . The dashed curves indicate the amplitude of the scattered wave obtained from the scalar theory single-slit approximation for each value of 2 b / λ . The single-slit data are normalized to the scaled theory value at ϕ = 0 .

Fig. 3
Fig. 3

Amplitude of the scattered field from the thin elliptic cylinder as a function of the angle ϕ in the Σ system with ρ = 5.2 m and 2 b / λ = 120 . The continuous curve is obtained from Eq. (33) and the dashed curve from the scalar theory single-slit approximation. The single-slit evaluation is normalized to | E z s / E o | at ϕ = 0 and 2 b / λ = 120 .

Fig. 4
Fig. 4

Amplitude of the scattered field from the thin elliptic cylinder in the Σ system. The solid curve is the data plotted in Fig. 3 transformed to the Σ system, and the solid points are obtained from Eq. (40) with ρ = 5.2 m and 2 b / λ = 120 .

Tables (1)

Tables Icon

Table 1 Values of the Parameters Used for the Evaluation of the Scattering of a Plane Wave Normally Incident on a Thin Elliptic Cylinder

Equations (40)

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

× E = i ω μ · H , × H = i ω ε · E ,
ε = [ ε o 0 0 0 ε o 0 0 0 ε o ] ,
μ = [ μ o 0 0 0 μ o 0 0 0 μ o ] .
= T ,
T = [ 1 / s x 0 0 0 1 / s y 0 0 0 1 / s z ] .
E = T E ,
H = T H ,
× E = i ω μ · H , × H = i ω ε · E ,
ε = [ ε x 0 0 0 ε y 0 0 0 ε z ] = ε o [ s y s z / s x 0 0 0 s z s x / s y 0 0 0 s x s y / s z ] ,
μ = [ μ x 0 0 0 μ y 0 0 0 μ z ] = μ o [ s y s z / s x 0 0 0 s z s x / s y 0 0 0 s x s y / s z ] .
x 2 a 2 + y 2 b 2 = 1 .
s x = 2 a a + b , s y = 2 b a + b , s z = 1 ,
x 2 + y 2 = R 2 , with     R = a + b 2 ,
E = ( 0 , 0 , E z ( x , y ) ) ,
E = ( 0 , 0 , E z ( x , y ) ) ,
1 μ y 2 E z x 2 + 1 μ x 2 E z y 2 + ω 2 ε z E z = 0 ,
E z = E o e i ( k x x + k y y ) ,
k = ω ε o s x s y { cos 2 ϕ o μ y + sin 2 ϕ o μ x } 1 / 2 .
k = k { ( cos ϕ o s x ) 2 + ( sin ϕ o s y ) 2 } 1 / 2 ,
ϕ o = tan 1 ( s x s y tan ϕ o ) , ϕ = tan 1 ( s x s y tan ϕ ) .
k = k g o g o o ,
g o = { cos 2 ϕ o s x 2 + sin 2 ϕ o s y 2 } 1 / 2 ,
g o o = { cos 2 ϕ o s x 4 + sin 2 ϕ o s y 4 } 1 / 2 ,
k x = k s x g o o cos ϕ o , k y = k s y g o o sin ϕ o .
x = cos ϕ o x + sin ϕ o y , y = sin ϕ o x + cos ϕ o y .
{ cos 2 ϕ o μ y + sin 2 ϕ o μ x } 2 E z x 2 + 2 sin ϕ o cos ϕ o { 1 μ x 1 μ y } 2 E z x y + { cos 2 ϕ o μ x + sin 2 ϕ o μ y } 2 E z y 2 + ω 2 ε z E z = 0 .
{ K 1 cos 2 ϕ + K 2 sin ϕ cos ϕ + K 3 sin 2 ϕ } 2 E z ρ 2 + 1 ρ 2 { 2 ( K 1 K 3 ) sin ϕ cos ϕ + K 2 ( sin 2 ϕ cos 2 ϕ ) } E z ϕ + 2 ρ { ( K 3 K 1 ) sin ϕ cos ϕ + K 2 ( cos 2 ϕ sin 2 ϕ ) } 2 E z ρ ϕ + 1 ρ { K 1 sin 2 ϕ 2 K 2 sin ϕ cos ϕ + K 3 cos 2 ϕ } E z ρ + 1 ρ 2 { K 1 sin 2 ϕ K 2 sin ϕ cos ϕ + K 3 cos 2 ϕ } 2 E z ϕ 2 + ω 2 ε z E z = 0 ,
K 1 = cos 2 ϕ o μ y + sin 2 ϕ o μ x ,
K 2 = 2 sin ϕ o cos ϕ o ( 1 μ x 1 μ y ) ,
K 3 = cos 2 ϕ o μ x + sin 2 ϕ o μ y .
1 ρ ( ρ E z ρ ) + 1 ρ 2 2 E z ϕ 2 + ω 2 c 2 E z = 0 ,
E z ( ρ , ϕ , t ) = E o n = + i n { J n ( k ρ ) J n ( k a ) H n ( 1 ) ( k a ) H n ( 1 ) ( k ρ ) } e i n ϕ e i ω t ,
E z s ( ρ , ϕ ) = E o n = + i n J n ( k R ) H n ( 1 ) ( k R ) H n ( 1 ) ( k ρ ) e i n ϕ .
ϕ 1 = ϕ ϕ o = tan 1 ( s y s x tan ( ϕ o + ϕ ) ) .
E z s ( ρ , ϕ ) = E o n = + i n J n ( k R ) H n ( 1 ) ( k R ) H n ( 1 ) ( k ρ ) g n g n o × ( cos n ϕ s x + i sin n ϕ s y ) ( cos n ϕ o s x i sin n ϕ o s y ) ,
g n = { ( cos n ϕ s x ) 2 + ( sin n ϕ s y ) 2 } 1 / 2 ,
g n o { ( cos n ϕ o s x ) 2 + ( sin n ϕ o s y ) 2 } 1 / 2 .
k = k { ( cos ϕ o s x ) 2 + ( sin ϕ o s y ) 2 } 1 / 2 ,
k ρ = k ρ g o g o o { cos 2 ϕ s x 2 + sin 2 ϕ s y 2 } 1 / 2 .
E z s ( ρ , ϕ ) = E o n = + i n J n ( k R ) H n ( 1 ) ( k R ) H n ( 1 ) ( k ρ ) e i n ( ϕ ϕ o ) ,

Metrics