Abstract

The accurate analysis of the electromagnetic field scattered by lenslets of dimensions of the order of the wavelength is considered. Assuming plane-wave illumination of the lenslet, a pair of coupled integral equations is derived, starting from the Stratton–Chu solution of Maxwell’s time-harmonic equations. These equations are solved rigorously (in a numerical sense) to obtain the lenslet-scattered field. For comparison, an approximate vector formula based on the aperture field is also derived and is related to the ubiquitous Fres-nel–Kirchhoff scalar diffraction formula. These analytical techniques are then applied to representative lenslets with diameters in the 2–100-wavelength range. The results demonstrate useful focusing characteristics even at 2–wavelength diameters and confirm that, depending on the accuracy required, the approximate vector formula based on the aperture field can be successfully employed for determining the lenslet-scattering characteristics.

© 1995 Optical Society of America

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References

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  12. L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from electrically large coated flat and curved strips: entire domain Galerkin’s formulation,” IEEE Trans. Antennas Propag. AP-35, 790–801 (1987).
    [CrossRef]
  13. M. R. Barclay, W. V. T. Rusch, “Moment-method analysis of large, axially symmetric reflector antennas using entire-domain functions,” IEEE Trans. Antennas Propag. 39, 491–496 (1991).
    [CrossRef]
  14. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), Sec. 3.5.
  15. J. R. Mautz, “Computer program for the Mie series solution for a sphere,” Tech. Rep. TR-77-12 (Syracuse University, Syracuse, N. Y., 1977).
  16. J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined field solutions for bodies of revolution,” Tech. Rep. TR-77-2 (Syracuse University, Syracuse, N. Y., 1977).
  17. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Sec. 8.3.
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 5.1.
  19. C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (Institute of Electrical and Electronics Engineers, New York, 1994).

1994 (1)

T. Nagata, T. Tanaka, K. Miyake, H. Kurotaki, S. Yokoyama, M. Koyanagi, “Micron-size optical waveguide for optoelectronic integrated circuit,” Jpn. J. Appl. Phys. 33, 822–826 (1994).
[CrossRef]

1991 (3)

1990 (1)

1987 (1)

L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from electrically large coated flat and curved strips: entire domain Galerkin’s formulation,” IEEE Trans. Antennas Propag. AP-35, 790–801 (1987).
[CrossRef]

1985 (2)

1982 (1)

1974 (1)

Baer, T. M.

Banno, J.

Barclay, M. R.

M. R. Barclay, W. V. T. Rusch, “Moment-method analysis of large, axially symmetric reflector antennas using entire-domain functions,” IEEE Trans. Antennas Propag. 39, 491–496 (1991).
[CrossRef]

Barnes, F. S.

Bellmann, R. H.

Borelli, N. F.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Sec. 8.3.

Cohen, L. G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 5.1.

Harrington, R. F.

J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined field solutions for bodies of revolution,” Tech. Rep. TR-77-2 (Syracuse University, Syracuse, N. Y., 1977).

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), Sec. 3.5.

J. R. Mautz, R. F. Harrington, “Electromagnetic scattering from a homogeneous body of revolution,” Tech. Rep. TR-77-10 (Syracuse University, Syracuse, N. Y., 1977).

Iga, K.

Jahns, J.

Kokubun, Y.

Koyanagi, M.

T. Nagata, T. Tanaka, K. Miyake, H. Kurotaki, S. Yokoyama, M. Koyanagi, “Micron-size optical waveguide for optoelectronic integrated circuit,” Jpn. J. Appl. Phys. 33, 822–826 (1994).
[CrossRef]

Kurotaki, H.

T. Nagata, T. Tanaka, K. Miyake, H. Kurotaki, S. Yokoyama, M. Koyanagi, “Micron-size optical waveguide for optoelectronic integrated circuit,” Jpn. J. Appl. Phys. 33, 822–826 (1994).
[CrossRef]

Lee, K. S.

Mautz, J. R.

J. R. Mautz, R. F. Harrington, “Electromagnetic scattering from a homogeneous body of revolution,” Tech. Rep. TR-77-10 (Syracuse University, Syracuse, N. Y., 1977).

J. R. Mautz, “Computer program for the Mie series solution for a sphere,” Tech. Rep. TR-77-12 (Syracuse University, Syracuse, N. Y., 1977).

J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined field solutions for bodies of revolution,” Tech. Rep. TR-77-2 (Syracuse University, Syracuse, N. Y., 1977).

Medgyesi-Mitschang, L. N.

L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from electrically large coated flat and curved strips: entire domain Galerkin’s formulation,” IEEE Trans. Antennas Propag. AP-35, 790–801 (1987).
[CrossRef]

Misawa, S.

Miyake, K.

T. Nagata, T. Tanaka, K. Miyake, H. Kurotaki, S. Yokoyama, M. Koyanagi, “Micron-size optical waveguide for optoelectronic integrated circuit,” Jpn. J. Appl. Phys. 33, 822–826 (1994).
[CrossRef]

Morgan, W. L.

Morse, D. L.

Nagata, T.

T. Nagata, T. Tanaka, K. Miyake, H. Kurotaki, S. Yokoyama, M. Koyanagi, “Micron-size optical waveguide for optoelectronic integrated circuit,” Jpn. J. Appl. Phys. 33, 822–826 (1994).
[CrossRef]

Nolscher, U.

Oikawa, M.

Putnam, J. M.

L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from electrically large coated flat and curved strips: entire domain Galerkin’s formulation,” IEEE Trans. Antennas Propag. AP-35, 790–801 (1987).
[CrossRef]

Riechert, P.

Rusch, W. V. T.

M. R. Barclay, W. V. T. Rusch, “Moment-method analysis of large, axially symmetric reflector antennas using entire-domain functions,” IEEE Trans. Antennas Propag. 39, 491–496 (1991).
[CrossRef]

Schneider, M. V.

Snyder, J. J.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 8.14.

Streibl, N.

Tai, C. T.

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (Institute of Electrical and Electronics Engineers, New York, 1994).

Tanaka, T.

T. Nagata, T. Tanaka, K. Miyake, H. Kurotaki, S. Yokoyama, M. Koyanagi, “Micron-size optical waveguide for optoelectronic integrated circuit,” Jpn. J. Appl. Phys. 33, 822–826 (1994).
[CrossRef]

Walker, S. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Sec. 8.3.

Yokoyama, S.

T. Nagata, T. Tanaka, K. Miyake, H. Kurotaki, S. Yokoyama, M. Koyanagi, “Micron-size optical waveguide for optoelectronic integrated circuit,” Jpn. J. Appl. Phys. 33, 822–826 (1994).
[CrossRef]

Appl. Opt. (7)

IEEE Trans. Antennas Propag. (2)

L. N. Medgyesi-Mitschang, J. M. Putnam, “Electromagnetic scattering from electrically large coated flat and curved strips: entire domain Galerkin’s formulation,” IEEE Trans. Antennas Propag. AP-35, 790–801 (1987).
[CrossRef]

M. R. Barclay, W. V. T. Rusch, “Moment-method analysis of large, axially symmetric reflector antennas using entire-domain functions,” IEEE Trans. Antennas Propag. 39, 491–496 (1991).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Nagata, T. Tanaka, K. Miyake, H. Kurotaki, S. Yokoyama, M. Koyanagi, “Micron-size optical waveguide for optoelectronic integrated circuit,” Jpn. J. Appl. Phys. 33, 822–826 (1994).
[CrossRef]

Other (9)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 8.14.

S. Silver, ed., Microwave Antenna Theory and Design (Peregrinus, London, 1986), Sec. 3.9.

J. R. Mautz, R. F. Harrington, “Electromagnetic scattering from a homogeneous body of revolution,” Tech. Rep. TR-77-10 (Syracuse University, Syracuse, N. Y., 1977).

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), Sec. 3.5.

J. R. Mautz, “Computer program for the Mie series solution for a sphere,” Tech. Rep. TR-77-12 (Syracuse University, Syracuse, N. Y., 1977).

J. R. Mautz, R. F. Harrington, “H-field, E-field, and combined field solutions for bodies of revolution,” Tech. Rep. TR-77-2 (Syracuse University, Syracuse, N. Y., 1977).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Sec. 8.3.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 5.1.

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (Institute of Electrical and Electronics Engineers, New York, 1994).

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Figures (12)

Fig. 1
Fig. 1

Geometry for the Stratton–Chu formulation.

Fig. 2
Fig. 2

Stratton–Chu formulation adapted to the lenslet geometry, (a) Original geometry, (b) equivalent exterior geometry, (c) equivalent interior geometry.

Fig. 3
Fig. 3

Surface of revolution geometry.

Fig. 4
Fig. 4

Geometry for lenslet AF formulation.

Fig. 5
Fig. 5

Spherical lenslet geometry.

Fig. 6
Fig. 6

Electric field scattered by a D = 5λ ball lenslet (a) along the z axis, (b) along the transverse focal plane.

Fig. 7
Fig. 7

Electric field scattered by a double-convex lenslet where D = 100λ and Δ0 = 5.22λ. (a), (b) Same as in Fig. 6.

Fig. 8
Fig. 8

Electric-field scattered by a plano-convex lenslet where D = 50λ and Δ0 = 1.33λ. (a), (b) Same as in Fig. 6.

Fig. 9
Fig. 9

Electric field scattered by a plano-convex lenslet where D = 50λ and Δ0 = 10.00λ. (a), (b) Same as in Fig. 6.

Fig. 10
Fig. 10

Electric field scattered by a plano-convex lenslet where D = 50λ and Δ0 = 25.00λ. (a), (b) Same as in Fig. 6.

Fig. 11
Fig. 11

Electric field scattered by a plano-convex lenslet where D = 5λ and Δ0 = 1.00λ. (a), (b) Same as in Fig. 6.

Fig. 12
Fig. 12

Electric field scattered by a plano-convex lenslet where D = 2λ and Δ0 = 0.40λ. (a), (b) Same as in Fig. 6.

Equations (34)

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E ( P ) = 1 4 π V [ j ω μ ψ J ( r ) + M ( r ) × ψ ρ e ( r ) ψ ] d V + 1 4 π S 1 + . . . + S n { j ω μ ψ [ n ̂ × H ( r ) ] + [ n ̂ × E ( r ) ] × ψ + [ n ̂ E ( r ) ] ψ } d S ,
H ( P ) = 1 4 π V [ j ω ψ M ( r ) J ( r ) × ψ ρ m ( r ) μ ψ ] d V + 1 4 π S 1 + . . . + S n { j ω ψ [ n ̂ × E ( r ) ] + [ n ̂ × H ( r ) ] × ψ + n ̂ H ( r ) ψ } d S ,
ψ = ψ ( r , r ) = exp ( jkR ) R , R = r r , R = | r r | ,
E inc ( P ) = x ̂ E 0 exp ( jkz ) , H inc ( P ) = ŷ E 0 η exp ( jkz ) ,
lim r r { [ r × H ( P ) ] + μ E ( P ) } = 0 .
E ( P ) = E inc ( P ) + E scat ( P ) ,
E scat ( P ) = 1 4 π S L { j ω μ ψ [ n ̂ × H ( r ) ] + [ n ̂ × E ( r ) ] ψ + [ n ̂ E ( r ) ] ψ } d S L .
J eq = n ̂ × H S L ,
M eq = n ̂ × E S L ,
ρ eq 0 = n ̂ E S L .
J eq = j ω ρ eq .
( J eq ψ ) = ( J eq ) ψ ,
ψ = ( j k + 1 / R ) ψ R ̂ ,
E scat ( P ) = j k η 0 4 π S L { J eq ( r ) [ 1 1 k 2 R ( j k + 1 R ) ] [ J eq ( r ) R ̂ ] R ̂ [ 1 3 k 2 R ( j k + 1 R ) ] + M eq ( r ) × R ̂ η 0 ( 1 + 1 jkR ) } ψ d S L ,
n ̂ × [ E inc ( r ) + E scat ( r ) ] S L = n ̂ × E diel ( r ) S L ,
n ̂ × [ H inc ( r ) + H scat ( r ) ] S L = n ̂ × H diel ( r ) S L .
e ( F ) = j ω μ 4 π S L F ( r ) ψ ( r , r ) d S L + j 4 π ω s S L [ s F ( r ) ] ψ ( r , r ) d S L ,
m ( F ) = s × 1 4 π S L F ( r ) ψ ( r , r ) d S L ,
n ̂ × E inc ( r ) = E ( J , M ) = [ e int ( J ) + m int ( M ) ] [ e ext ( J ) m ext ( M ) ] ,
n ̂ × H inc ( r ) = M ( J , M ) = [ e int ( M ) n d 2 m int ( J ) ] [ e ext ( M ) n 0 2 + m ext ( J ) ] .
F ( r ) = F ( , ϕ ) = n = N N m = 0 M [ F n m 1 ρ sin ( m π L ) ̂ + F n m ϕ cos ( m π L ) ϕ ̂ ] exp ( j n ϕ ) ,
W ( r ) = W ( , ϕ ) = n = N N m = 0 M [ 1 ρ sin ( m π L ) ̂ + cos ( m π L ) ϕ ̂ ] exp ( j n ϕ ) ,
Ω , W = S L Ω ( , ϕ ) W * ( , ϕ ) d S ,
[ E inc , W H inc , W ] = [ E ( J , M ) , W H ( J , M ) , W ] ,
E ( P ) = 1 4 π S A { j ω μ ψ [ n ̂ × H ( P ) ] + [ n ̂ × E ( P ) ] ψ + [ n ̂ E ( P ) ] ψ } d S A .
E S A ( ρ , ϕ ) t ( ρ , ϕ ) E inc z = Δ 0 , H S A ( ρ , ϕ ) t ( ρ , ϕ ) H inc z = Δ 0 ,
t ( ρ , ϕ ) = { exp [ j k Φ ( ρ ) ] ρ D / 2 exp ( j k Δ 0 ) otherwise .
Φ ( ρ ) = n Δ 0 ( n 1 ) ( R a { 1 [ 1 ( ρ / R a ) 2 ] 1 / 2 } + R b { 1 [ 1 ( ρ / R b ) 2 ] 1 / 2 } ) ,
E ( P ) j 2 λ S A L E 0 exp { j k [ Φ 0 ( ρ ) Δ 0 ] } × Λ ψ d S A L + j 2 λ S A S E 0 Λ ψ d S A S ,
Λ = x ̂ [ 1 + ( 1 + 1 jkR ) ( R ̂ ) ] [ 1 + ( 1 + 1 jkR ) ( x ̂ R ̂ ) ] .
j 2 λ S A S E 0 Λ ψ d S A S = E inc ( P ) j 2 λ S A L E 0 Λ ψ d S A L ,
E ( P ) E inc ( P ) + j 2 λ 0 D / 2 0 2 π E 0 exp { j k [ Φ ( ρ ) Δ 0 ] } × Λ ψ ρ d ρ d ϕ .
Λ x ̂ [ 1 + ( R ̂ ) ]
I dB = 10 log ( E E * E inc E inc * ) ,

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