Abstract

The fine structure of the annular images of circular gratings is analyzed in terms of diffraction patterns of axicon pairs. Each image arises substantially from only one converging and one diverging axicon of equal deflection angle. Single main lobe, symmetric double main lobe, and various intermediate asymmetric double lobe structures are obtained depending on the phase relationship and strengths of the two axicon beams, which in turn depend on the design of the circular grating. Approximate expressions are derived for the intensity distribution and energy content of the symmetric single and double lobe images.

© 1974 Optical Society of America

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References

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  1. D. Tichenor, R. N. Bracewell, J. Opt. Soc. Am. 63, 1620 (1973).
    [CrossRef]
  2. R. N. Bracewell, A. R. Thompson, Astrophys. J. 182, 77 (1973).
    [CrossRef]
  3. J. Dyson, Proc. R. Soc. Lond. A248, 93 (1958).
  4. J. K. McLeod, J. Opt. Soc. Am. 44, 592 (1959).
    [CrossRef]
  5. S. Fujiwaru, J. Opt. Soc. Am. 57, 287 (1962).
    [CrossRef]
  6. J. W. Lit, F. Brennan, J. Opt. Soc. Am. 60, 370 (1970).
    [CrossRef]
  7. M. Abramovitz, L. A. Stegun, Eds., Handbook of Mathematical Functions (National Bureau of Standards, Appl. Math. Series 55, Washington, D.C., 1964), p. 321.
  8. The Fourier coefficients for gratings consisting of alternating opaque and transparent zones are b±m = ±(mπ)−1 for odd m; b±m = 0 for even m > 2; and bo = 1/2.
  9. A. Boivin, Theorie et Calcul des Figures de Diffraction de Revolution (Gauthier Villars, Les Presses de l’Université Laval Quebec, 1964).
  10. A. Fedotowsky, K. Lehovec, Appl. Opt. 13, Dec. (1974).
    [PubMed]

1974 (1)

A. Fedotowsky, K. Lehovec, Appl. Opt. 13, Dec. (1974).
[PubMed]

1973 (2)

D. Tichenor, R. N. Bracewell, J. Opt. Soc. Am. 63, 1620 (1973).
[CrossRef]

R. N. Bracewell, A. R. Thompson, Astrophys. J. 182, 77 (1973).
[CrossRef]

1970 (1)

1962 (1)

S. Fujiwaru, J. Opt. Soc. Am. 57, 287 (1962).
[CrossRef]

1959 (1)

1958 (1)

J. Dyson, Proc. R. Soc. Lond. A248, 93 (1958).

Boivin, A.

A. Boivin, Theorie et Calcul des Figures de Diffraction de Revolution (Gauthier Villars, Les Presses de l’Université Laval Quebec, 1964).

Bracewell, R. N.

D. Tichenor, R. N. Bracewell, J. Opt. Soc. Am. 63, 1620 (1973).
[CrossRef]

R. N. Bracewell, A. R. Thompson, Astrophys. J. 182, 77 (1973).
[CrossRef]

Brennan, F.

Dyson, J.

J. Dyson, Proc. R. Soc. Lond. A248, 93 (1958).

Fedotowsky, A.

A. Fedotowsky, K. Lehovec, Appl. Opt. 13, Dec. (1974).
[PubMed]

Fujiwaru, S.

S. Fujiwaru, J. Opt. Soc. Am. 57, 287 (1962).
[CrossRef]

Lehovec, K.

A. Fedotowsky, K. Lehovec, Appl. Opt. 13, Dec. (1974).
[PubMed]

Lit, J. W.

McLeod, J. K.

Thompson, A. R.

R. N. Bracewell, A. R. Thompson, Astrophys. J. 182, 77 (1973).
[CrossRef]

Tichenor, D.

Appl. Opt. (1)

A. Fedotowsky, K. Lehovec, Appl. Opt. 13, Dec. (1974).
[PubMed]

Astrophys. J. (1)

R. N. Bracewell, A. R. Thompson, Astrophys. J. 182, 77 (1973).
[CrossRef]

J. Opt. Soc. Am. (4)

Proc. R. Soc. Lond. (1)

J. Dyson, Proc. R. Soc. Lond. A248, 93 (1958).

Other (3)

M. Abramovitz, L. A. Stegun, Eds., Handbook of Mathematical Functions (National Bureau of Standards, Appl. Math. Series 55, Washington, D.C., 1964), p. 321.

The Fourier coefficients for gratings consisting of alternating opaque and transparent zones are b±m = ±(mπ)−1 for odd m; b±m = 0 for even m > 2; and bo = 1/2.

A. Boivin, Theorie et Calcul des Figures de Diffraction de Revolution (Gauthier Villars, Les Presses de l’Université Laval Quebec, 1964).

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Figures (6)

Fig. 1
Fig. 1

Coordinates in the aperture and image planes.

Fig. 2
Fig. 2

Equivalent axicons of radial periodic filters. The converging and diverging axicons of strength bm and bm combine to generate the mth order image |Vm|2 = |Umα + Umα|2 located at the deflection angle αm = mRo/f = mα.

Fig. 3
Fig. 3

The function (C12 + S12)/2 characterizes the ring image of a single axicon. The function (C1S1)2/2 describes the shape of all images (m = 1,3 …) of a circular binary phase grating of inner disk radius ro equal to the zone width Δr = π/zo. The functions C12 and S12 are asymptotic forms to the ring images m = 1,5,9 … and m = 3,7,11 …, respectively, of a circular binary phase grating of inner disk radius ro = 3/4 Δr.

Fig. 4
Fig. 4

Percentage energy flux H ( z ¯ ) within an annulus of width 2 z ¯ with respect to the total energy in the image order under consideration. The function Hc applies to images of intensity distribution C12, and Hs applies to the intensity distribution S12 shown in Fig. 3.

Fig. 5
Fig. 5

The first four ring images of the eight-zone binary circular grating sgn cos(zorπ/4).

Fig. 6
Fig. 6

First and third order images,) |V(z)|2, of the ten-zone circular binary grating sgn cos(zorπ/4) and the asymptotic forms C12 and S12. Abscissa is z ¯ = z m z 0 where mzo is the center of the mth ring image.

Tables (1)

Tables Icon

Table I Fine Structure of the Diffraction Images of Binary Circular Gratings for Four Special Values of the Phase Difference −π < γmπ of Corresponding Axicon Pairs

Equations (26)

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r = ρ / a , z = 2 π a R / λ f ,
F ( r ) = b o + m > 0 ( b m e i m z o r + b m e i m z o r ) .
U α ( z ) = 2 0 1 e i α r J o ( z r ) r d r ,
U α | α | 1 / 2 ( 1 + i ) E 1 * ( z ¯ ) / 2 ,
z ¯ = z | α | = z m z o ,
E 1 = C 1 + i S 1 ,
C 1 S 1 ( z ¯ ) = ( 2 π ) 1 / 2 0 1 cos sin ( z ¯ r ) r 1 / 2 d r = z ¯ 1 [ ± ( 2 π ) 1 / 2 sin cos ( z ¯ ) 1 2 S o C o ( z ¯ ) ]
C o ( z ) S o ( z ) = ( 2 π ) 1 / 2 0 1 cos sin ( z r ) r 1 / 2 d r
E 1 2 = C 1 2 + S 1 2 ,
V m ( z ¯ ) = b m U m z o + b m U m z o * = e i π / 4 ( b m E 1 * b m E 1 ) ( m z o ) 1 / 2 = e i π / 4 [ ( b m i b m ) C 1 i ( b m + i b m ) S 1 ] ( m z o ) 1 / 2 ,
e i γ m = b m / b m ,
V m ( z ¯ ) = 2 | b m | e i γ m / 2 ( m z o ) 1 / 2 · [ C 1 e i γ m / 2 cos ( γ m / 2 + π / 4 ) + S 1 e + i γ m / 2 cos ( γ m / 2 π / 4 ) ] .
| V m ( z ¯ ) | 2 = 4 | b m | 2 ( m z o ) 1 C 1 2 ( z ¯ ) S 1 2 ( z ¯ ) .
| V m ( 0 ) 2 | = 32 | b m | 2 [ 9 π m z o ] 1 = 1.14 | b m | 2 ( m z o ) 1 ,
| V m ( ± 0.7 π ) | 2 = 0.78 | b m | 2 ( m z o ) 1 .
H m ( z ¯ ) = 1 2 m z o z ¯ m z o + z ¯ | V m ( z ) | 2 z d z .
H m ( z ¯ ) 2 | b m | 2 H C ( z ¯ ) H S ( z ¯ ) ,
H C ( z ¯ ) H S ( z ¯ ) = z ¯ + z ¯ | C 1 | S 1 ( z ¯ ) | 2 d z ¯ .
H m ( ) = 2 | b m | 2 .
= ± 2 / ( m π ) , m = 1,3 b ± m = 0 , m = 0,2,4 ;
γ m = π ( 1 2 m r o / Δ r ) .
| b ± m | = π / 2 Δ ρ
γ m = 4 π m ρ o / 2 Δ ρ
U α ( z ) 2 ( 2 π z ) 1 / 2 0 1 e i α r cos ( z r π / 4 ) r 1 / 2 d r ,
U α ( z ) = ( 2 z ) 1 / 2 [ ( 1 + i ) E 1 * ( z ¯ ) + ( 1 i ) E 1 ( z ¯ + 2 α ) ]
e i α r cos ( z r π / 4 ) = [ ( 1 i ) e i ( 2 α r + z ¯ r ) + ( 1 + i ) e i z ¯ r ] / ( 2 2 ) .

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