Abstract

Systems of cascaded circular apertures may be built to focus electromagnetic waves if the difference (d) between the Fresnel numbers of any two apertures pertaining to two points (A and B) is an even number. With a point source at A, the spherical incident wave will be focused to B. In general, the irradiance at B (principal focus) is roughly proportional to (n±1)2 where n is the number of apertures; the plus sign is used when the Fresnel numbers are odd integers, and minus when even. A system of a few apertures with d = 0 can focus waves over a wide range of frequencies, though the actual irradiance at the principal focus is a function of frequency. A condition to maximize the irradiance at the principal focus has been found. Theoretically such systems have been analyzed by the boundary-diffraction-wave theory generalized by Miyamoto and Wolf. Analytical expressions for the diffraction wave amplitude have been obtained for axial and off-axial points. when the iteration method of Fox and Li is also applicable, the two theoretical results agree very well with each other. Experimental data obtained agree well with the theoretical results.

© 1969 Optical Society of America

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References

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  1. M. De, J. W. Y. Lit, and R. Tremblay, Appl. Opt. 7, 483 (1968).
    [CrossRef] [PubMed]
  2. R. Tremblay and M. De, Appl. Phys. Letters 9, 136 (1966).
    [CrossRef]
  3. M. De, R. Tremblay, and J. W. Y. Lit, J. Opt. Soc. Am. 54, 1437 (1966).
  4. T. Young, A Course of Lectures an Natural Philosophy and Mechanical Arts (London, 1807).
    [CrossRef]
  5. A. Rubinowicz, Ann. Physik 4, 53, 257 (1917).
  6. K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 615 (1962).
    [CrossRef]
  7. K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 626 (1962).
    [CrossRef]
  8. The subscript n in Un(j)(Q) means the number of diffractions the wave has undergone, while the superscript (j) indicates the last aperture which diffracts the wave and from which the wave propagates to the point Q.
  9. Since the scalar wave theory is the basis of the calculations, the point Q must be far enough from the edge so that the theory is applicable. Also, Q must be off the axis by a distance great enough so that the contribution from the points with stationary phase represents reasonably well the boundary wave.
  10. E. T. Copson, A symptotic Expansions (Cambridge University Press, Cambridge, 1965).
    [CrossRef]
  11. E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 58, 720 (1968).
  12. A. Boivin, Théorie et calcul des figures de diffraction de révolution (Les Presses de l’Université Laval, Québec, 1964).
  13. E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).
  14. G. Toraldo di Francia, Atti. Fond. Giorgio Ronchi Contrib. Inst. Natl. Optica 6, 3 (1951).
  15. R. W. Wood, Phil. Mag. 45, 511 (1898).
  16. J. L. Soret, Arch. Sci. Phys. Nature 52, 320 (1875).
  17. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Natl. Bur. Stds., Washington, D. C., 1964; Dover Publications, Inc., New York, 1965), p. 361.
  18. G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1958).
  19. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, Inc., New York, 1964).
  20. A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).
    [CrossRef]

1968 (2)

M. De, J. W. Y. Lit, and R. Tremblay, Appl. Opt. 7, 483 (1968).
[CrossRef] [PubMed]

E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 58, 720 (1968).

1966 (2)

R. Tremblay and M. De, Appl. Phys. Letters 9, 136 (1966).
[CrossRef]

M. De, R. Tremblay, and J. W. Y. Lit, J. Opt. Soc. Am. 54, 1437 (1966).

1962 (2)

1961 (1)

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).
[CrossRef]

1953 (1)

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

1951 (1)

G. Toraldo di Francia, Atti. Fond. Giorgio Ronchi Contrib. Inst. Natl. Optica 6, 3 (1951).

1917 (1)

A. Rubinowicz, Ann. Physik 4, 53, 257 (1917).

1898 (1)

R. W. Wood, Phil. Mag. 45, 511 (1898).

1875 (1)

J. L. Soret, Arch. Sci. Phys. Nature 52, 320 (1875).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Natl. Bur. Stds., Washington, D. C., 1964; Dover Publications, Inc., New York, 1965), p. 361.

Boivin, A.

A. Boivin, Théorie et calcul des figures de diffraction de révolution (Les Presses de l’Université Laval, Québec, 1964).

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, Inc., New York, 1964).

Copson, E. T.

E. T. Copson, A symptotic Expansions (Cambridge University Press, Cambridge, 1965).
[CrossRef]

De, M.

Fox, A. G.

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).
[CrossRef]

Li, T.

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

Lit, J. W. Y.

Marchand, E. W.

E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 58, 720 (1968).

Miyamoto, K.

Rubinowicz, A.

A. Rubinowicz, Ann. Physik 4, 53, 257 (1917).

Soret, J. L.

J. L. Soret, Arch. Sci. Phys. Nature 52, 320 (1875).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Natl. Bur. Stds., Washington, D. C., 1964; Dover Publications, Inc., New York, 1965), p. 361.

Toraldo di Francia, G.

G. Toraldo di Francia, Atti. Fond. Giorgio Ronchi Contrib. Inst. Natl. Optica 6, 3 (1951).

Tremblay, R.

Watson, G. N.

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1958).

Wolf, E.

E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 58, 720 (1968).

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 626 (1962).
[CrossRef]

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 615 (1962).
[CrossRef]

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, Inc., New York, 1964).

Wood, R. W.

R. W. Wood, Phil. Mag. 45, 511 (1898).

Young, T.

T. Young, A Course of Lectures an Natural Philosophy and Mechanical Arts (London, 1807).
[CrossRef]

Ann. Physik (1)

A. Rubinowicz, Ann. Physik 4, 53, 257 (1917).

Appl. Opt. (1)

Appl. Phys. Letters (1)

R. Tremblay and M. De, Appl. Phys. Letters 9, 136 (1966).
[CrossRef]

Arch. Sci. Phys. Nature (1)

J. L. Soret, Arch. Sci. Phys. Nature 52, 320 (1875).

Atti. Fond. Giorgio Ronchi Contrib. Inst. Natl. Optica (1)

G. Toraldo di Francia, Atti. Fond. Giorgio Ronchi Contrib. Inst. Natl. Optica 6, 3 (1951).

Bell System Tech. J. (1)

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).
[CrossRef]

J. Opt. Soc. Am. (4)

Phil. Mag. (1)

R. W. Wood, Phil. Mag. 45, 511 (1898).

Proc. Phys. Soc. (London) (1)

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

Other (8)

A. Boivin, Théorie et calcul des figures de diffraction de révolution (Les Presses de l’Université Laval, Québec, 1964).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Natl. Bur. Stds., Washington, D. C., 1964; Dover Publications, Inc., New York, 1965), p. 361.

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1958).

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, Inc., New York, 1964).

The subscript n in Un(j)(Q) means the number of diffractions the wave has undergone, while the superscript (j) indicates the last aperture which diffracts the wave and from which the wave propagates to the point Q.

Since the scalar wave theory is the basis of the calculations, the point Q must be far enough from the edge so that the theory is applicable. Also, Q must be off the axis by a distance great enough so that the contribution from the points with stationary phase represents reasonably well the boundary wave.

E. T. Copson, A symptotic Expansions (Cambridge University Press, Cambridge, 1965).
[CrossRef]

T. Young, A Course of Lectures an Natural Philosophy and Mechanical Arts (London, 1807).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Arrangement of apertures.

Fig. 2
Fig. 2

Geometry of diffraction by a circular aperture.

Fig. 3
Fig. 3

Geometry of diffraction by two circular apertures.

Fig. 4
Fig. 4

Irradiance distribution in principal focal plane of system with N = 5, M = 2, C(−1, 0, 1). Values have been normalized with respect to maximum. Full Curve: iteration method. Crosses: boundary-diffraction-wave method. Circles: experimental data.

Fig. 5
Fig. 5

Irradiance distribution in principal focal plane of system with N = 5, M = 4, C(−5, −4, ⋯, 0, ⋯5). Values have been normalized with respect to maximum. Full curve: theoretical results. Circles: experimental data.

Fig. 6
Fig. 6

Irradiance distribution along axis of system with N = 5, M = 2, C(−1, 0, 1). Values have been normalized with respect to maximum. Full curve: iteration method. The crosses denote results given by the boundary-diffraction-wave method when only the geometric-optical wave and the singly diffracted boundary waves are taken into account. When the doubly diffracted waves are also included, the results are shown by the circles.

Fig. 7
Fig. 7

Irradiance at principal focus of system with N = 5, M = 4. Values have been normalized with respect to the irradiance produced by free space propagation. Full curve: theoretical results. Crosses: experimental data.

Fig. 8
Fig. 8

Irradiances at principal focus for various positions of apertures of system with N = 5, a0 = 20λ. Curve A: n = 2, C(0) fixed midway between source and focus; the position and radius of C(1) varied to give different values of Δ. Curve B: n = 3; C(0) fixed; C(1) fixed at position corresponding to maximum point of curve A; C(2) varied. Curve C: Positions of C(0,1) are the same as those of curve B; C(2) fixed at position corresponding to maximum point of curve B; C(3) varied. All irradiances are normalized with respect to the irradiance produced by free-space propagation. Full curves are theoretical results. Circles are experimental data.

Fig. 9
Fig. 9

Diffraction pattern produced by system with N = 9, M = 8, n = 16, a0 = 2907λ for λ = 5461 Å. A point source of white light was used.

Fig. 10
Fig. 10

Illustrating the construction of a system of circular apertures with equal radii.

Fig. 11
Fig. 11

Theoretical results of irradiance at principal focus as a function of frequency. Values have been normalized with respect to the irradiance produced by free-space propagation. Curve A: ellipsoidal system with radii and spacings of apertures satisfying Eq. (33). Curve B: system with apertures of equal radii. Both systems have L = 318.75λ, a0 = 20λ, n = 5.

Equations (58)

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U ( r ) = A ( r ) exp [ i k ϕ ( r ) ] ,
ϕ ( r ) = 1
A ( r ) / A ( r ) k = 2 π / λ ,
U K ( P ) = U B ( P ) + U G ( P ) ,
U B ( P ) = 1 4 π Γ U ( Q ) exp ( i k ρ ) ρ r × ϱ 1 - r · ϱ · l d l ,
U ( S ) = U G ( S ) + j = 1 n U 1 ( j ) ( S ) + j = 2 n U 2 ( j ) ( S ) + multiply diffracted waves ,
U 1 ( P ) = A a 1 4 π 0 2 π exp [ i k ( r 1 + ρ ) ] r 1 ρ r 1 × ϱ 1 - r 1 · ϱ · l d ϕ ,
r 1 × ϱ 1 - r 1 · ϱ · l = z 1 x cos ϕ - a 1 z r 1 ρ + a 1 2 - a 1 x cos ϕ - z 1 s 1 ,
U 1 ( P ) = - a 1 2 z exp { i k ( r 1 + ρ 1 ) } / 2 r 1 ρ 1 ( r 1 ρ 1 + a 1 2 - z 1 s 1 ) .
r 1 ρ 1 - z 1 s 1 = a 1 2 ( r 1 2 + ρ 1 2 ) / 2 r 1 ρ 1 .
U 1 ( P ) = - exp { i k ( r 1 + ρ 1 ) } / z .
λ a 1 s 1 ,
ρ 1 = s 1 + ( a 1 2 + x 2 - 2 a 1 x cos ϕ ) / 2 s 1 .
exp ( - i z cos ϕ ) = J 0 ( z ) + 2 m = 1 ( - i ) m J m ( z ) cos ( m ϕ ) ,
U 1 ( P ) = C [ C 0 J 0 ( v ) - i C 1 J 1 ( v ) - C 2 E ] ,
E = β - 1 m = 0 m ( - i ) m α m J m ( v ) ,
C = - a 1 exp { i k [ r 1 + s 1 + ( a 1 2 + x 2 ) / 2 s 1 ] } 2 r 1 s 1 { r 1 s 1 [ 1 + ( a 1 2 + x 2 ) / 2 s 1 2 ] + a 1 2 - z 1 s 1 } ,
C 0 = ( B 1 + C 1 ) / B 3 ,
C 1 = B 2 / B 3 ,
C 2 = B 0 + C 0 ,
B 0 = - a 1 z { 1 - ( a 1 2 + x 2 ) / 2 s 1 2 } ,
B 1 = z 1 x { 1 - ( a 1 2 + x 2 ) / 2 s 1 2 } - z a 1 2 x / s 1 2 ,
B 2 = z 1 a 1 x 2 / s 1 2 ,
B 3 = a 1 x ( 1 + r 1 / s 1 ) / { r 1 s 1 [ 1 + ( a 1 2 + x 2 ) / 2 s 1 2 ] + a 1 2 - z 1 s 1 } ,
α = ( 1 - β ) / β 3 ,
β = ( 1 - β 3 2 ) 1 2             ( positive root ) ,
v = k a 1 x / s 1 ,
U 1 ( P ) = - ( 1 / z ) exp { i k [ z + a 1 2 ( 1 / z 1 + 1 / s 1 ) / 2 + x 2 / 2 s 1 ] } × { U 0 ( u , v ) - i U 1 ( u , v ) } ,
U 1 ( P ) = - ( 1 / z ) exp { i k [ z + a 1 2 ( 1 / z 1 + 1 / s 1 ) / 2 + x 2 / 2 s 1 ] } × { V 2 ( u , v ) + i V 1 ( u , v ) } ,
u = k z 1 x 2 / s 1 z .
U n ( u , v ) + U n + 2 ( u , v ) = ( u / v ) n J n ( v ) ,
U 1 ( P ) = ( x z 1 - a 1 z ) ( a 1 λ / d 1 x ) 1 2 exp { i k ( r 1 + d 1 + λ / 8 ) } 4 π r 1 ( r 1 d 1 - a 1 x + a 1 2 - z 1 t 1 ) .
U 2 ( P ) = 1 4 π Γ U 1 ( Q 2 ) exp ( i k ρ ) ρ · p ˆ 2 × ρ ˆ 1 - p ˆ 2 · ρ ˆ l d l ,
U 2 ( P ) = U 1 Q s ( Q 2 ) · a 2 { a 1 s 2 - a 2 ( t 1 + s 2 ) } × exp ( i k ρ 2 ) / 2 ρ 2 { d 1 ρ 2 + a 2 ( a 2 - a 1 ) - t 1 s 2 } ,
U 2 ( P ) = C ( C 0 J 0 ( v ) - i C 1 J 1 ( v ) - C 2 E ) ,
E = β - 1 m = 0 m ( - i ) m α m J m ( v ) ,
C = - a 2 2 s 2 · exp { i k [ s 2 + ( a 2 2 + x 2 ) / 2 s 2 ] } d 1 s 2 { 1 + ( a 2 2 + x 2 ) / 2 s 2 2 } + a 2 ( a 2 - a 1 ) - t 1 s 2 × U 1 Q s ( Q 2 ) ,
C 0 = ( B 1 + C 1 ) / B 3 ,
C 1 = B 2 / B 3 ,
C 2 = B 0 + C 0 ,
B 0 = { 1 - ( a 2 2 + x 2 ) / 2 s 2 2 } { s 2 ( a 2 - a 1 ) + t 1 a 2 } ,
B 1 = ( a 2 x / s 2 2 ) { s 2 ( a 2 - a 1 ) + t 1 a 2 } - t 1 x { 1 - ( a 2 2 + x 2 ) / 2 s 2 2 } ,
B 2 = - t 1 a 2 x 2 / s 2 2 ,
B 3 = ( d 1 a 2 / s 2 + a 2 - a 1 ) x d 1 s 2 { 1 + ( a 2 2 + x 2 ) / 2 s 2 2 } + a 2 ( a 2 - a 1 ) - t 1 s 2 ,
α = ( 1 - β ) / B 3 ,
β = ( 1 - B 3 2 ) 1 2             ( positive root ) ,
v = k a 2 x / s 2 .
U ( P ) = U G ( P ) + j = 1 n U 1 ( j ) ( P ) + j = 2 n U 2 ( j ) ( P ) ,
U ( S ) = 1 L exp ( i k L ) - 1 L j = 1 n exp { i k ( r j + ρ j ) } + j = 2 n A j exp { i k ( r j - 1 + d j - 1 + ρ j + λ / 8 ) } ,
A j = ( a j z j - 1 - a j - 1 z j ) { a j - 1 s j - a j ( t j - 1 + s j ) } ( λ a j - 1 a j / d j - 1 ) 1 2 8 π r j - 1 ρ j { r j - 1 d j - 1 - a j - 1 ( a j - a j - 1 ) z j - 1 t j - 1 } { d j - 1 ρ j + a j ( a j - a j - 1 ) - t j - 1 s j } , t j - 1 = z j - z j - 1 ,             s j = L - z j ,             d j - 1 = { ( a j - a j - 1 ) 2 + t j - 1 2 } 1 2 ,
r j + ρ j = L + N λ / 2 ,
r j - 1 + d j - 1 + ρ j = L + N λ / 2 + Δ .
A j = A .
U ( S ) = exp ( i k L ) / L - exp { i k ( L + N λ / 2 ) } × [ n / L - ( n - 1 ) A exp { i k ( λ / 8 + Δ ) } ] .
Δ = a 0 2 ( N - M ) / 2 L ( N M ) 1 2
A = { λ · M 1 2 / 2 L ( N 1 2 - M 1 2 ) } 1 2 / π a 0 ,
N = 2 p + 1             ( p = 0 , 1 , 2 , ) ,
Δ + λ / 8 = ( 2 q + 1 ) λ / 2             ( q = 0 , 1 , 2 , ) .