Abstract

The Fraunhofer diffraction patterns of the triangle, trapezoid, and hexagon are calculated and displayed in moiré plots. The diffraction pattern of the general polygonal aperture with a finite number of vertices can be calculated in terms of elementary functions.

© 1974 Optical Society of America

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References

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  1. J. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice–Hall, Englewood Cliffs, N. J., 1972), Ch. 2.3.
  2. G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart and Winston, New York, 1968), Sec. 4.4.
  3. A. Sommerfeld, Optics (Academic, New York, 1954), p. 233ff.
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), p. 86.
  5. Reference 4, p. 9.
  6. M. Stecher, Am. J. Phys. 32, 247 (1964).
    [Crossref]
  7. G. Oster, M. Wasserman, and C. Zwerling, J. Opt. Soc. Am. 54, 169 (1964).
    [Crossref]
  8. H. Takasaki, Appl. Opt. 9, 1467 (1970).
    [Crossref] [PubMed]

1970 (1)

1964 (2)

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart and Winston, New York, 1968), Sec. 4.4.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), p. 86.

Meyer-Arendt, J.

J. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice–Hall, Englewood Cliffs, N. J., 1972), Ch. 2.3.

Oster, G.

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1954), p. 233ff.

Stecher, M.

M. Stecher, Am. J. Phys. 32, 247 (1964).
[Crossref]

Takasaki, H.

Wasserman, M.

Zwerling, C.

Am. J. Phys. (1)

M. Stecher, Am. J. Phys. 32, 247 (1964).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Other (5)

J. Meyer-Arendt, Introduction to Classical and Modern Optics (Prentice–Hall, Englewood Cliffs, N. J., 1972), Ch. 2.3.

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart and Winston, New York, 1968), Sec. 4.4.

A. Sommerfeld, Optics (Academic, New York, 1954), p. 233ff.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), p. 86.

Reference 4, p. 9.

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Figures (7)

Fig. 1
Fig. 1

Geometry for the transforming lens. The Fourier plane is always at the image plane of the illuminating source, in this case a point source at infinity.

Fig. 2
Fig. 2

Isosceles-triangular aperture with half-angle α. The object function O(x,y) is unity within the aperture, zero elsewhere.

Fig. 3
Fig. 3

Calculated and photographed diffraction patterns for the triangle of Fig. 2, with α = 30°.

Fig. 4
Fig. 4

Trapezoidal aperture.

Fig. 5
Fig. 5

Calculated and photographed diffraction patterns for the trapezoid of Fig. 4, with α = 7.5°. The light fans from the parallel top and bottom edges overlap completely, with resulting sinc(y) = sin(πy)/πy behavior.

Fig. 6
Fig. 6

Regular hexagonal aperture, shown divided into two complementary trapezoids. The trapezoid of Fig. 4 is translated and stretched as required to correspond to the comparative dimensions shown here.

Fig. 7
Fig. 7

Calculated and photographed diffraction patterns for the regular hexagon of Fig. 6. The three pairs of parallel edges produce six fans, each with the sinc function behavior characteristic of a finite-width slit.

Equations (22)

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A ( ξ , η ) = 1 i f λ exp [ i π f λ ( 1 - d f ) ( ξ 2 + η 2 ) ] × Aperture 0 ( x , y ) exp [ - 2 π i ( x ξ + y η ) / ( f λ ) ] d x d y ,
X = ( 2 π / f λ ) ξ , Y = ( 2 π / f λ ) η ;
A ( X , Y ) = 0 ( x , y ) exp [ - i ( x X + y Y ) ] d x d y .
A ( X , Y ) = y = - 1 0 exp ( - i Y y ) x = y tan α - y tan α exp ( - i X x ) d x .
A ( X , Y ) = ( 1 / X ) × y = - 1 0 [ exp ( 2 i y T - ) - exp ( - 2 i y T + ) ] d y .
T ± = ( X tan α ± Y ) / 2.
A ( X , Y ) = ( 1 / X ) × [ exp ( i T + ) sin T + T + - exp ( - i T - ) sin T - T - ] .
A ( X , Y ) = exp [ - i ( T + - T - ) ] sin ( T + - T - ) T + T - + exp [ i ( T + + T 0 ) ] sin ( T + + T 0 ) T + T 0 - exp [ i ( T - + T 0 ) ] sin ( T - + T 0 ) T - T 0 ,
B ( X , Y ) = ( 1 / X ) [ exp ( i X ) sin t + t + - exp ( - i X ) sin t - t - ] ,
t ± = X tan α ± Y .
F [ O ( x , y ) ] = B ( X , Y ) ,
F [ 0 ( x , y ) ] = B ( X , Y ) exp ( - i Y a ) B ( X , Y ) .
F [ O ( x , y ) ] = B ( X , Y ) ,
F [ O ( x , c y ) ] = ( 1 / c ) B ( X , Y / c ) .
F [ O ( x , y ) ] = B ( X , Y ) ,
C ( X , Y ) = ( 1 / X ) [ sin ( v + ) sin u - u - + sin ( v - ) sin u + u + ] ,
u ± = t ± / 3 , v ± = X ± 2 Y / 3 .
B ( x ) = 1 + cos ( 2 π x / s ) .
B ( x , y ) = 1 + cos [ 2 π x / s - k f ( x , y ) ] ,
T ( x ) = 1 + cos ( 2 π x / s ) ,
I T = B ( x , y ) · T ( x ) = 1 + cos ( 2 π x / s ) + cos [ 2 π x / s - k f ( x , y ) ] + 1 2 cos [ 4 π x / s - k f ( x , y ) ] + 1 2 cos [ k f ( x , y ) ] ,
f = tanh [ 1 - 1 2 log ( I / I 0 ) ] .