Abstract

The necessary and sufficient conditions are derived in order that an infinite plane object, illuminated by a plane monochromatic wave of normal incidence, images itself without the aid of lenses or other optical accessories. This involves a solution of the reduced wave equation which does not satisfy the Sommerfeld radiation condition. The solution is obtained by requiring a geometrical-optics limiting condition as the wavelength λ goes to zero. Two cases of self-imaging are considered. The first case, called weak, deals with the faithful imaging of objects whose spatial frequencies are all much smaller than the (1/λ) value of the illuminating source. The conditions for this case demand that the two-dimensional Fourier spectrum of the object lies on the circles of a Fresnel zone plate. The second case, called strong, deals with the faithful imaging of objects for spatial frequencies up to the natural cutoff of 1/λ. Both doubly- and singly-periodic and nonperiodic objects are considered. For periodic objects the results are shown to agree well with the experimental and theoretical work to date, the latter of which has always employed the Fresnel–Kirchhoff diffraction integral with the parabolic approximation appropriate to Fresnel diffraction.

© 1967 Optical Society of America

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References

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  1. H. F. Talbot, Phil. Mag. and J. Sci. London 9, 401 (1836).
  2. Lord Rayleigh, Phil. Mag. (5),  2, 196 (1881).
  3. M. Wolfke, Ann. Physik 40, 194 (1913).
    [Crossref]
  4. F. Zernike, Physik. Z. 36, 848 (1935).
  5. H. H. Hopkins, Proc. Roy. Soc. (London), Ser. A:  217, 408 (1953).
    [Crossref]
  6. J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London), Ser. B:  70, 4861957).
    [Crossref]
  7. John T. Winthrop and C. R. Worthington, J. Opt. Soc. Am. 55, 373 (1965).
    [Crossref]
  8. Max Born and Emil Wolf, Principles of Optics (Pergamon Press Ltd., Oxford, 1959), p. 379.
  9. Arnold Sommerfeld, Partial Differential Equations in Physics (Academic Press Inc., New York, 1949), p. 189.
  10. See Ref. 9, p. 190.
  11. D. Gabor, J. Inst. Elec. Engrs. (London) 93, 429 (1946).
  12. See Ref. 2, p. 203.
  13. D. Gabor, in Progress in Optics, I, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), p. 137.
  14. H. Gamo, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1964), p. 193.
  15. Arnold Sommerfeld, Optics (Academic Press Inc., New York, 1964), p. 209.
  16. Reference 15, p. 219.
  17. I. M. Ge’Fand and G. E. Shilov, Generalized Functions, Vol. I, Properties and Operations (Academic Press Inc., New York, 1964).

1965 (1)

1957 (1)

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London), Ser. B:  70, 4861957).
[Crossref]

1953 (1)

H. H. Hopkins, Proc. Roy. Soc. (London), Ser. A:  217, 408 (1953).
[Crossref]

1946 (1)

D. Gabor, J. Inst. Elec. Engrs. (London) 93, 429 (1946).

1935 (1)

F. Zernike, Physik. Z. 36, 848 (1935).

1913 (1)

M. Wolfke, Ann. Physik 40, 194 (1913).
[Crossref]

1881 (1)

Lord Rayleigh, Phil. Mag. (5),  2, 196 (1881).

1836 (1)

H. F. Talbot, Phil. Mag. and J. Sci. London 9, 401 (1836).

Born, Max

Max Born and Emil Wolf, Principles of Optics (Pergamon Press Ltd., Oxford, 1959), p. 379.

Cowley, J. M.

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London), Ser. B:  70, 4861957).
[Crossref]

Gabor, D.

D. Gabor, J. Inst. Elec. Engrs. (London) 93, 429 (1946).

D. Gabor, in Progress in Optics, I, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), p. 137.

Gamo, H.

H. Gamo, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1964), p. 193.

Ge’Fand, I. M.

I. M. Ge’Fand and G. E. Shilov, Generalized Functions, Vol. I, Properties and Operations (Academic Press Inc., New York, 1964).

Hopkins, H. H.

H. H. Hopkins, Proc. Roy. Soc. (London), Ser. A:  217, 408 (1953).
[Crossref]

Moodie, A. F.

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London), Ser. B:  70, 4861957).
[Crossref]

Rayleigh, Lord

Lord Rayleigh, Phil. Mag. (5),  2, 196 (1881).

Shilov, G. E.

I. M. Ge’Fand and G. E. Shilov, Generalized Functions, Vol. I, Properties and Operations (Academic Press Inc., New York, 1964).

Sommerfeld, Arnold

Arnold Sommerfeld, Optics (Academic Press Inc., New York, 1964), p. 209.

Arnold Sommerfeld, Partial Differential Equations in Physics (Academic Press Inc., New York, 1949), p. 189.

Talbot, H. F.

H. F. Talbot, Phil. Mag. and J. Sci. London 9, 401 (1836).

Winthrop, John T.

Wolf, Emil

Max Born and Emil Wolf, Principles of Optics (Pergamon Press Ltd., Oxford, 1959), p. 379.

Wolfke, M.

M. Wolfke, Ann. Physik 40, 194 (1913).
[Crossref]

Worthington, C. R.

Zernike, F.

F. Zernike, Physik. Z. 36, 848 (1935).

Ann. Physik (1)

M. Wolfke, Ann. Physik 40, 194 (1913).
[Crossref]

J. Inst. Elec. Engrs. (London) (1)

D. Gabor, J. Inst. Elec. Engrs. (London) 93, 429 (1946).

J. Opt. Soc. Am. (1)

Phil. Mag. (1)

Lord Rayleigh, Phil. Mag. (5),  2, 196 (1881).

Phil. Mag. and J. Sci. London (1)

H. F. Talbot, Phil. Mag. and J. Sci. London 9, 401 (1836).

Physik. Z. (1)

F. Zernike, Physik. Z. 36, 848 (1935).

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

J. M. Cowley and A. F. Moodie, Proc. Phys. Soc. (London), Ser. B:  70, 4861957).
[Crossref]

Proc. Roy. Soc. (London) (1)

H. H. Hopkins, Proc. Roy. Soc. (London), Ser. A:  217, 408 (1953).
[Crossref]

Other (9)

Max Born and Emil Wolf, Principles of Optics (Pergamon Press Ltd., Oxford, 1959), p. 379.

Arnold Sommerfeld, Partial Differential Equations in Physics (Academic Press Inc., New York, 1949), p. 189.

See Ref. 9, p. 190.

See Ref. 2, p. 203.

D. Gabor, in Progress in Optics, I, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), p. 137.

H. Gamo, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1964), p. 193.

Arnold Sommerfeld, Optics (Academic Press Inc., New York, 1964), p. 209.

Reference 15, p. 219.

I. M. Ge’Fand and G. E. Shilov, Generalized Functions, Vol. I, Properties and Operations (Academic Press Inc., New York, 1964).

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Figures (6)

F. 1
F. 1

Object of infinite aperture in x, y, plane.

F. 2
F. 2

Phase shift of a spatial frequency ξ0.

F. 3
F. 3

A self-imaging, nonperiodic object.

F. 4
F. 4

Two-dimensional direct and reciprocal lattices.

F. 5
F. 5

The lattice, zone-plate relation for weak imaging.

F. 6
F. 6

Strong imaging conditions.

Equations (40)

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f ( x ) = R 2 F ( ξ ) exp ( 2 π i x ξ ) d ξ ,
φ ( x , z ) = R 2 Φ ( ξ , z ) exp ( 2 π i x ξ ) d ξ .
[ Δ ( 1 / c 2 ) ( 2 / t 2 ) ] φ e i ω t = 0
R 2 | ξ | 2 | Φ ( ξ , z ) | d ξ and R 2 | 2 Φ ( ξ , z ) z 2 | d ξ
R 2 [ Φ ( ξ , z ) K 2 ( ξ ) + Φ ( ξ , z ) ] exp ( 2 π i x ξ ) d ξ = 0
Φ ( ξ , z ) = { A ( ξ ) cos K ( ξ ) z + B ( ξ ) sin K ( ξ ) z for | ξ | 1 / λ C ( ξ ) e i K ( ξ ) z + D ( ξ ) e i K ( ξ ) z for | ξ | > 1 / λ ,
F ( ξ ) = { A ( ξ ) for | ξ | 1 / λ C ( ξ ) for | ξ | > 1 / λ .
φ ( x , z ) = R 2 F ( ξ ) exp ( i K ( ξ ) z ) exp ( 2 π i x ξ ) d ξ .
K ( ξ ) = 2 π ( λ 2 ξ 2 ) 1 2 k [ 1 1 2 ( λ ξ ) 2 ] .
φ ( x , d ) = α f ( x ) for all x ,
F ( ξ ) exp [ i K ( ξ ) d ] = α F ( ξ ) for all ξ .
exp i k [ 1 1 2 ( λ ξ ) 2 ] d = α = e 2 π i μ ,
( d / λ ) [ 1 1 2 ( λ ξ ) 2 ] = μ + m ( ξ ) ,
d / λ = μ + m ( 0 ) ,
( d / 2 λ ) ( λ ξ ) 2 = m ( ξ ) ,
| ξ | = ( 2 / d λ ) 1 2 [ m ( ξ ) ] 1 2 .
( λ λ ) ( d d ) = 2 .
F ( ξ ) = a δ ( ξ O ) + b δ [ | ξ | ( 2 / λ δ ) 1 2 ] ,
f ( x ) = a + b 2 π ( 2 λ d ) 1 2 J 0 [ 2 π | x | ( 2 λ d ) 1 2 ] ,
f ( x ) = n F n exp ( 2 π i x b n ) ,
φ ( x , z ) = n F n exp [ i K ( b n ) z ] exp ( 2 π i x b n ) ,
| b n | = ( 2 / λ d ) 1 2 [ m ( n ) ] 1 2
f ( x ) = f ( x , y ) = f ( x + n a , y )
| n / a | = ( 2 / λ d ) 1 2 [ m ( n ) ] 1 2
( d / λ ) [ 1 ( λ ξ ) 2 ] 1 2 = μ + m ( ξ ) .
ξ 2 + ( m ( ξ ) d 1 λ ) 2 = ( 1 λ ) 2 ,
d = [ ( x y ) 2 + R 2 ] 1 2 R [ 1 + ( x y ) 2 / 2 R 2 ] ,
φ ( x , z ) = Λ [ F ( ξ ) cos K ( ξ ) z + B ( ξ ) sin K ( ξ ) z ] exp ( 2 π i x ξ ) d ξ + Λ c F ( ξ ) exp [ i K ( ξ ) z ] exp ( 2 π i x ξ ) d ξ ,
B ( ξ 0 ) = i F ( ξ 0 ) .
2 b 1 b 2 = α 2 [ m ( 1 , 1 ) m ( 1 , 0 ) m ( 0 , 1 ) ] ,
b n 2 = n 1 2 b 1 2 + 2 n 1 n 2 b 1 b 2 + n 2 2 b 2 2 = α 2 { n 1 2 m ( 1 , 0 ) + n 2 2 m ( 0 , 1 ) + n 1 n 2 [ m ( 1 , 1 ) m ( 1 , 0 ) m ( 0 , 1 ) ] } = α 2 m ( n ) ,
| b 2 | / | a 1 | = | b 1 | / | a 2 | .
b n + ν 2 = α 2 m ( n + ν ) = ( b n + b ν ) 2 = α 2 [ m ( n ) + m ( ν ) ] + 2 b n b ν .
2 b n b ν = α 2 [ m ( n + ν ) m ( n ) m ( ν ) ] = α 2 M ( n , ν ) .
M ( n , ν ) = M ( n , ν 1 e 1 + ν 2 e 2 ) = ν 1 M ( n , e 1 ) + ν 2 M ( n , e 2 ) = m ( n ) ν ,
m ( n ) = [ M ( n , e 1 ) , M ( n , e 2 ) ] .
2 b n b ν = α 2 M ( n , ν ) = α 2 m ( n ) ν = α 2 a m b ν
b ν [ b n ( α 2 / 2 ) a m ] = 0 for all ν .
( 2 / α 2 ) b n = a m ( n ) for all n ,
b n 2 = ( α 2 / 2 ) a m ( n ) b n = ( α 2 / 2 ) [ m ( n ) n ] for all n