Abstract

Investigation of electromagnetic diffraction at any distance from an infinite slit is presented for slit widths 0.5 wavelength and wider. Plane-wave incidence of arbitrary two-dimensional angle in a plane perpendicular to the edges of the slit is used. Infinite conductivity is assumed. The contribution of this work is in the near-field and medium-slit-width regions, where no numerical results have been given. One obvious application is high-resolution, contact or near-contact, printing.

© 1972 Optical Society of America

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References

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  1. C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
    [Crossref]
  2. J. B. Keller and E. B. Hansen, Acta Phys. Polon. 27, 217 (1965).
  3. J. S. Asvestas and R. E. Kleinman, in Electromagnetic and Acoustic Scattering by Simple Shapes, edited by J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi (North–Holland, Amsterdam, 1969), Ch. 4.
  4. P. M. Morse and P. J. Rubenstein, Phys. Rev. 54, 895 (1938).
    [Crossref]
  5. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), Eq. (8.3.7).
  6. Reference 5, Eq. (11.5.22).
  7. G. Bekefi, J. Appl. Phys. 24, 1123 (1953).
    [Crossref]
  8. H. P. Hsu, Scientific Report No. 5, Case Institute of Technology, Cleveland, Ohio, 1959.
  9. H. S. Tan, Australian J. Phys. 21, 35 (1968).
    [Crossref]

1968 (1)

H. S. Tan, Australian J. Phys. 21, 35 (1968).
[Crossref]

1965 (1)

J. B. Keller and E. B. Hansen, Acta Phys. Polon. 27, 217 (1965).

1954 (1)

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
[Crossref]

1953 (1)

G. Bekefi, J. Appl. Phys. 24, 1123 (1953).
[Crossref]

1938 (1)

P. M. Morse and P. J. Rubenstein, Phys. Rev. 54, 895 (1938).
[Crossref]

Asvestas, J. S.

J. S. Asvestas and R. E. Kleinman, in Electromagnetic and Acoustic Scattering by Simple Shapes, edited by J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi (North–Holland, Amsterdam, 1969), Ch. 4.

Bekefi, G.

G. Bekefi, J. Appl. Phys. 24, 1123 (1953).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), Eq. (8.3.7).

Bouwkamp, C. J.

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
[Crossref]

Hansen, E. B.

J. B. Keller and E. B. Hansen, Acta Phys. Polon. 27, 217 (1965).

Hsu, H. P.

H. P. Hsu, Scientific Report No. 5, Case Institute of Technology, Cleveland, Ohio, 1959.

Keller, J. B.

J. B. Keller and E. B. Hansen, Acta Phys. Polon. 27, 217 (1965).

Kleinman, R. E.

J. S. Asvestas and R. E. Kleinman, in Electromagnetic and Acoustic Scattering by Simple Shapes, edited by J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi (North–Holland, Amsterdam, 1969), Ch. 4.

Morse, P. M.

P. M. Morse and P. J. Rubenstein, Phys. Rev. 54, 895 (1938).
[Crossref]

Rubenstein, P. J.

P. M. Morse and P. J. Rubenstein, Phys. Rev. 54, 895 (1938).
[Crossref]

Tan, H. S.

H. S. Tan, Australian J. Phys. 21, 35 (1968).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), Eq. (8.3.7).

Acta Phys. Polon. (1)

J. B. Keller and E. B. Hansen, Acta Phys. Polon. 27, 217 (1965).

Australian J. Phys. (1)

H. S. Tan, Australian J. Phys. 21, 35 (1968).
[Crossref]

J. Appl. Phys. (1)

G. Bekefi, J. Appl. Phys. 24, 1123 (1953).
[Crossref]

Phys. Rev. (1)

P. M. Morse and P. J. Rubenstein, Phys. Rev. 54, 895 (1938).
[Crossref]

Rept. Progr. Phys. (1)

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
[Crossref]

Other (4)

H. P. Hsu, Scientific Report No. 5, Case Institute of Technology, Cleveland, Ohio, 1959.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), Eq. (8.3.7).

Reference 5, Eq. (11.5.22).

J. S. Asvestas and R. E. Kleinman, in Electromagnetic and Acoustic Scattering by Simple Shapes, edited by J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi (North–Holland, Amsterdam, 1969), Ch. 4.

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Figures (12)

Fig. 1
Fig. 1

The slit, incident beam, and coordinates.

Fig. 2
Fig. 2

The relations of r, ψ, and α.

Fig. 3
Fig. 3

Square of absolute amplitude of field components and magnitude of the Poynting vector due to the plane-wave diffraction of a three-wavelength slit. The three curves in each block correspond to distances of 0, 3, and 10 wavelengths from the slit. Normal incidence was used.

Fig. 4
Fig. 4

Same as Fig. 3, except that the angle of incidence is 45°.

Fig. 5
Fig. 5

Three-dimensional plot showing the evolution of the diffracted wave from the aperture plane. The slit is located at z = 0 and is centered at x = 0.

Fig. 6
Fig. 6

The TE counterpart of Fig. 5.

Fig. 7
Fig. 7

Same as Fig. 5 but with 45° angle of incidence.

Fig. 8
Fig. 8

The TE counterpart of Fig. 7.

Fig. 9
Fig. 9

The TE diffraction of a one-wavelength slit. Normal incidence.

Fig. 10
Fig. 10

Same as Fig. 9, but with 45° angle of incidence.

Fig. 11
Fig. 11

Ten-wavelength slit. Normal TE incidence.

Fig. 12
Fig. 12

Ten-wavelength slit. 45° TE incidence.

Tables (1)

Tables Icon

Table I Comparison of the TE aperture fields given by half-plane synthesis and Mathieu-function expansion (Ref. 8).

Equations (34)

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E y d ( r , ψ , α ) = ( 1 / π ) e i ( k r + 3 π / 4 ) { G ( u ) + G ( v ) } ,
G ( a ) e - i a 2 a e i t 2 d t ,
u ( 2 k r ) 1 2 cos [ ( ψ - α ) / 2 ] ,
v - ( 2 k r ) 1 2 cos [ ( ψ + α ) / 2 ] .
E y = E y i + C p E y p d + C n E y n d ,
E y i = e i k ( x sin θ + z cos θ )
C p = e i k x w p sin θ
C n = e i k x w n sin θ
E y n 1 d = C p E y p d ( w , π , α p ) E y n d ( r n , ψ n , π ) .
E y p 1 d = C n E y n d ( w , π , α n ) E y p d ( r p , ψ p , π ) .
E y n 2 d = E y p 1 d ( w , π , α p ) E y n d ( r n , ψ n , π ) = C n E y n d ( w , π , α n ) E y p d ( w , π , π ) E y n d ( r n , ψ n , π )
E y p 2 d = C p E y p d ( w , π , α p ) E y n d ( w , π , π ) E y p d ( r p , ψ p , π ) .
E y = E y i + C p E y p d + C n E y n d + E y p 1 d + E y n 1 d + E y p 2 d + E y n 2 d .
E y i = e i k ( x sin θ + z cos θ ) ,
E y p d = ( B / 2 ) e i k r p [ G ( u p ) + G ( v p ) ] ,
E y p 1 d = C n B 2 e i k ( r p + w ) G [ ( 2 k w ) 1 2 sin ( α n / 2 ) ] × G [ ( 2 k r p ) 1 2 sin ( ψ p / 2 ) ] ,
E y p 2 d = C p B 3 e i k ( r p + 2 w ) G [ ( 2 k w ) 1 2 ] G [ ( 2 k w ) 1 2 sin ( α p / 2 ) ] × G [ ( 2 k r p ) 1 2 sin ( ψ p / 2 ) ] ,
H x i = - E y i cos θ ,
H x p d = ( B / 2 ) e i k r p { - [ G ( u p ) - G ( v p ) ] sin α p + i ( 2 / k r p ) 1 2 sin ( α p / 2 ) cos ( ψ p / 2 ) } ,
H x p 1 d = i C n [ B 2 / ( 2 k r p ) - 1 2 ] e i k ( r p + w ) × G [ ( 2 k w ) 1 2 sin ( α n / 2 ) ] cos ( ψ p / 2 ) ,
H x p 2 d = i C p [ B 3 / ( 2 k r p ) - 1 2 ] e i k ( r p + 2 w ) G [ ( 2 k w ) 1 2 ] × G [ ( 2 k w ) 1 2 sin ( α p / 2 ) ] cos ( ψ p / 2 ) ,
H z i = E y i sin θ ,
H z p d = ( - B / 2 ) e i k r p { [ G ( u p ) + G ( v p ) ] cos α p + i ( 2 / k r p ) 1 2 sin ( α p / 2 ) sin ( ψ p / 2 ) } ,
H z p 1 d = C n B 2 e i k ( r p + w ) G [ ( k w ) 1 2 sin ( α n / 2 ) ] × G [ ( 2 k r p ) 1 2 sin ( ψ p / 2 ) ] ,
H z p 2 d = C p B 3 e i k ( r p + 2 w ) G [ ( 2 k w ) 1 2 ] G [ ( 2 k w ) 1 2 sin ( α p / 2 ) ] × G [ ( 2 k r p ) 1 2 sin ( ψ p / 2 ) ] ,
C p e i ( k w / 2 ) sin θ ,
C n e - i ( k w / 2 ) sin θ ,
B i ( 2 / π ) e i π / 4 .
H y i = e i k ( x sin θ + z cos θ ) ,
H y p d = ( B / 2 ) e i k r p [ G ( u p ) - G ( v p ) ] ,
E x i = H y i cos θ ,
E x p d = ( B / 2 ) e i k r p { [ G ( u p ) + G ( v p ) ] sin α p - i ( 2 / k r p ) 1 2 cos ( α p / 2 ) sin ( ψ p / 2 ) } ,
E z i = - H y i sin θ ,
E z p d = ( B / 2 ) e i k r p { [ G ( u p ) - G ( v p ) ] cos α p - i ( 2 / k r p ) 1 2 cos ( α p / 2 ) cos ( ψ p / 2 ) } .