Abstract

Most solutions for electromagnetic diffraction by a circular aperture in a perfectly conducting plane screen are for an incident homogeneous (propagating) plane wave. When the aperture is electrically small (dimensions small compared to the wavelength), the well-known transmission coefficient behaves as the fourth power of the diameter/wavelength. We consider the case in which the incident field is an inhomogeneous (evanescent) plane wave. Numerical calculations for the electrically small circular aperture show that the transmission coefficient for an inhomogeneous plane wave can be substantially greater than for a homogeneous plane wave at the same frequency. This observation may be helpful in explaining the increased transmission recently reported for electrically small apertures in plane screens with modifications. The numerical calculations for the electrically small aperture are in agreement with results from approximate analytical expressions that are based on the equivalent electric and magnetic dipole moments for the electrically small complementary disk.

© 2004 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Chap. VIII.
  2. J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, Washington, D.C., 1987), Chap. 14.
  3. G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970), Vol. 2, Chap. 7.
  4. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
    [CrossRef]
  5. G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge U. Press, Cambridge, UK, 1997), Chap. 7.
  6. T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
    [CrossRef]
  7. A. Degiron, H. J. Lezec, W. L. Barnes, T. W. Ebbesen, “Effects of hole depth on light transmission through subwavelength hole arrays,” Appl. Phys. Lett. 81, 4327–4329 (2002).
    [CrossRef]
  8. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
    [CrossRef] [PubMed]
  9. A. P. Hibbins, J. R. Sambles, C. R. Lawrence, “Gratingless enhanced microwave transmission through a subwavelength aperture in a thick metal plate,” Appl. Phys. Lett. 81, 4661–4663 (2002).
    [CrossRef]
  10. G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge U. Press, Cambridge, UK, 1997), Chap. 4.
  11. FEKO suite 3.2.2, available from EM Software & Systems-SA (Pty) Ltd, P.O. Box 1354, Stellenbosch 7599 South Africa.
  12. W. H. Eggimann, “Higher-order evaluation of electromagnetic diffraction by circular disks,” IRE Trans. Microwave Theory Tech. 9, 408–418 (1961).
    [CrossRef]

2002

A. Degiron, H. J. Lezec, W. L. Barnes, T. W. Ebbesen, “Effects of hole depth on light transmission through subwavelength hole arrays,” Appl. Phys. Lett. 81, 4327–4329 (2002).
[CrossRef]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, “Gratingless enhanced microwave transmission through a subwavelength aperture in a thick metal plate,” Appl. Phys. Lett. 81, 4661–4663 (2002).
[CrossRef]

2001

1961

W. H. Eggimann, “Higher-order evaluation of electromagnetic diffraction by circular disks,” IRE Trans. Microwave Theory Tech. 9, 408–418 (1961).
[CrossRef]

1944

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

Barnes, W. L.

A. Degiron, H. J. Lezec, W. L. Barnes, T. W. Ebbesen, “Effects of hole depth on light transmission through subwavelength hole arrays,” Appl. Phys. Lett. 81, 4327–4329 (2002).
[CrossRef]

Barrick, D. E.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970), Vol. 2, Chap. 7.

Bethe, H. A.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Chap. VIII.

Bowman, J. J.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, Washington, D.C., 1987), Chap. 14.

Degiron, A.

A. Degiron, H. J. Lezec, W. L. Barnes, T. W. Ebbesen, “Effects of hole depth on light transmission through subwavelength hole arrays,” Appl. Phys. Lett. 81, 4327–4329 (2002).
[CrossRef]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

Devaux, E.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

Ebbesen, T. W.

A. Degiron, H. J. Lezec, W. L. Barnes, T. W. Ebbesen, “Effects of hole depth on light transmission through subwavelength hole arrays,” Appl. Phys. Lett. 81, 4327–4329 (2002).
[CrossRef]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[CrossRef]

Eggimann, W. H.

W. H. Eggimann, “Higher-order evaluation of electromagnetic diffraction by circular disks,” IRE Trans. Microwave Theory Tech. 9, 408–418 (1961).
[CrossRef]

Garcia-Vidal, F. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

Hibbins, A. P.

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, “Gratingless enhanced microwave transmission through a subwavelength aperture in a thick metal plate,” Appl. Phys. Lett. 81, 4661–4663 (2002).
[CrossRef]

Krichbaum, C. K.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970), Vol. 2, Chap. 7.

Lawrence, C. R.

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, “Gratingless enhanced microwave transmission through a subwavelength aperture in a thick metal plate,” Appl. Phys. Lett. 81, 4661–4663 (2002).
[CrossRef]

Lezec, H. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

A. Degiron, H. J. Lezec, W. L. Barnes, T. W. Ebbesen, “Effects of hole depth on light transmission through subwavelength hole arrays,” Appl. Phys. Lett. 81, 4327–4329 (2002).
[CrossRef]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[CrossRef]

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[CrossRef]

Martin-Moreno, L.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

Pellerin, K. M.

Ruck, G. T.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970), Vol. 2, Chap. 7.

Sambles, J. R.

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, “Gratingless enhanced microwave transmission through a subwavelength aperture in a thick metal plate,” Appl. Phys. Lett. 81, 4661–4663 (2002).
[CrossRef]

Senior, T. B. A.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, Washington, D.C., 1987), Chap. 14.

Smith, G. S.

G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge U. Press, Cambridge, UK, 1997), Chap. 7.

G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge U. Press, Cambridge, UK, 1997), Chap. 4.

Stuart, W. D.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970), Vol. 2, Chap. 7.

Thio, T.

Uslenghi, P. L. E.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, Washington, D.C., 1987), Chap. 14.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Chap. VIII.

Appl. Phys. Lett.

A. Degiron, H. J. Lezec, W. L. Barnes, T. W. Ebbesen, “Effects of hole depth on light transmission through subwavelength hole arrays,” Appl. Phys. Lett. 81, 4327–4329 (2002).
[CrossRef]

A. P. Hibbins, J. R. Sambles, C. R. Lawrence, “Gratingless enhanced microwave transmission through a subwavelength aperture in a thick metal plate,” Appl. Phys. Lett. 81, 4661–4663 (2002).
[CrossRef]

IRE Trans. Microwave Theory Tech.

W. H. Eggimann, “Higher-order evaluation of electromagnetic diffraction by circular disks,” IRE Trans. Microwave Theory Tech. 9, 408–418 (1961).
[CrossRef]

Opt. Lett.

Phys. Rev.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

Science

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
[CrossRef] [PubMed]

Other

G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge U. Press, Cambridge, UK, 1997), Chap. 4.

FEKO suite 3.2.2, available from EM Software & Systems-SA (Pty) Ltd, P.O. Box 1354, Stellenbosch 7599 South Africa.

G. S. Smith, An Introduction to Classical Electromagnetic Radiation (Cambridge U. Press, Cambridge, UK, 1997), Chap. 7.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Chap. VIII.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, Washington, D.C., 1987), Chap. 14.

G. T. Ruck, D. E. Barrick, W. D. Stuart, C. K. Krichbaum, Radar Cross Section Handbook (Plenum, New York, 1970), Vol. 2, Chap. 7.

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Figures (5)

Fig. 1
Fig. 1

Schematic drawing showing the details for the diffraction of an incident plane wave by a circular aperture in a perfectly conducting plane screen, for a homogeneous TE plane wave.

Fig. 2
Fig. 2

Schematic drawing for the problem that is complementary to the one shown in Fig. 1: the scattering of an incident plane wave by a perfectly conducting circular disk, for a homogeneous TM plane wave.

Fig. 3
Fig. 3

Transmission coefficient for a circular aperture with k0d=0.063 (d/λ0=0.01) as a function of the wave number kxi for the incident TE plane wave. The gray area shows the region where the incident plane wave is homogeneous.

Fig. 4
Fig. 4

As for Fig. 3, but for the incident TM plane wave.

Fig. 5
Fig. 5

Transmission coefficient for a circular aperture with k0d=3.14 (d/λ0=0.5) as a function of the wave number kxi for the incident plane wave. (a) TE wave, (b) TM wave. The gray area shows the region where the incident plane wave is homogeneous.

Equations (29)

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

τ(ki)=totaltime-averagepowertransmittedthroughtheapertureA(time-averagepowerperunitareaoftheincidentplanewave),  
τ(θi)4(k0d)427π2cos2 θi,TE4(k0d)427π214sin2 θi+1,TM.
Eai,TE=E0exp(-jkir)y,
Hai,TE=E0η0-kzik0x+kxik0zexp(-jkir),
Hai,TM=-E0η0exp(-jkir)y,
Eai,TM=E0-kzik0x+kxik0zexp(-jkir).
ki=kxix+kziz,with kiki=k02,
Sai(t)=Re12Eai×Hai*=12η0 |E0|2exp[2 Im(ki)r]Rekik0,
kzi
=(k02-kxi2)1/2,homogeneouswave-j(kxi2-k02)1/2=-jαzi,inhomogeneouswave,
Eai=E0exp[-j(kxix+kziz)]y,TEE0-kzik0x+kxik0zexp[-j(kxix+kziz)],TM
Sai(t)
=12η0 |E0|2kik0,TEandTM,
Eai=E0exp(-αziz)exp(-jkxix)y,TEE0j αzik0x+kxik0zexp(-αziz)exp(-jkxix),TM
Sai(t)
=12η0exp(-2αziz)kxik0|E0|2x,TEandTM.
kxi=k0sin θi,kzi=k0cos θi.
Sai(t)=12η0kxik0|E0|2x,TEandTM.
τ(ki)=1η0ϕ=0πθ=0π/2|rEar|2sin θ dθdϕA|Sai(t)|.
Edi=-η0Hai,Hdi=1η0Eai.
σT(ki)=totaltime-averagepowerscatteredtime-averagepowerperunitareaoftheincidentplanewave=2η0ϕ=0πθ=0π/2|rEdsr|2sin θ dθdϕ|Sdi(t)|,
τ(ki)=σT(ki)2A.
pd=2d33 0(Exdix+Eydiy)x=0,y=0,z=0,
md=-d33 (Hzdiz)x=0,y=0,z=0.
σT(ki)=ck0412π0|pd|2+1c2 |md|2|Sdi(t)|;
pd=20d33Exdi+(k0d)212013Exdi-3k022Exdiz2x+Eydi+(k0d)212013Eydi-3k022Eydiz2-j2ω0Hzdixyx=0,y=0,z=0=20d331+(k0d)212016-3kxik02Exdix+1+(k0d)212016-3kxik02Eydi-η0(k0d)260kxik0Hzdiyx=0,y=0,z=0,
md=-d33Hzdi-(k0d)240×3Hzdi+1k022Hzdiz2zx=0,y=0,z=0=-d331-(k0d)2402+kxik02Hzdizx=0,y=0,z=0.
τTE(ki)=4(k0d)427π2Rekik0-1kzik021+4(k0d)215 ×1-316kxik02+4(k0d)42251-38kxik02+9256kxik04,
τTM(ki)=4(k0d)427π2Rekik0-11+14kxik02+4(k0d)2151-1332kxik02-364kxik04+4(k0d)42251-151256kxik02+17128kxik04+91024kxik06.

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