Abstract

General expressions for the intensity distribution in Fraunhofer N-slit diffraction patterns with finite sources both in the form of a uniformly radiating slit and disk have been obtained. By making use of the secondary maxima in the multiple slit patterns and their dependence upon source dimensions it is shown that multiple slits give a limited but useful advantage over double slits for the determination of the angular extent of distant sources. Results indicate an effective increase in resolution of 1/N, N>2. This is verified experimentally.

© 1952 Optical Society of America

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References

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  1. A. A. Michelson, Phil. Mag. 30, 1 (1890).
    [Crossref]
  2. A. A. Michelson and F. G. Pease, Astrophys. J. 53, 249 (1921).
    [Crossref]
  3. For example, Jenkins and White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950), second edition, chapter 17.
  4. Although this progressive disappearance of submaxima has probably been observed in many laboratories, it apparently was not mentioned in the literature until as late as 1938. After the present work was completed our attention was called to an article in the Journal of Science of Hirosima University, Series A,  14, 239–242, Sept.1950 by Takeyama, Kitahara, and Matuhayasi giving a preliminary, qualitative treatment of this problem with a slit source. In this article reference is made to D. Onoyama, Journal of Science of Hirosima University,  8, 81 (1938), who found that the number of secondary maxima diminished, one by one, as the breadth of the slit of the collimator was gradually increased.
  5. D. Jackson, Fourier Series and Orthogonal Polynomials (The Mathematical Association of America, 1941), pp. 33, 34.
  6. Gray, Mathews, and MacRoberts, A Treatise on Bessel Functions (Macmillan and Company, Ltd., London, England, 1931), p. 46.
  7. See reference 3, pp. 312–317.

1950 (1)

Although this progressive disappearance of submaxima has probably been observed in many laboratories, it apparently was not mentioned in the literature until as late as 1938. After the present work was completed our attention was called to an article in the Journal of Science of Hirosima University, Series A,  14, 239–242, Sept.1950 by Takeyama, Kitahara, and Matuhayasi giving a preliminary, qualitative treatment of this problem with a slit source. In this article reference is made to D. Onoyama, Journal of Science of Hirosima University,  8, 81 (1938), who found that the number of secondary maxima diminished, one by one, as the breadth of the slit of the collimator was gradually increased.

1921 (1)

A. A. Michelson and F. G. Pease, Astrophys. J. 53, 249 (1921).
[Crossref]

1890 (1)

A. A. Michelson, Phil. Mag. 30, 1 (1890).
[Crossref]

Gray,

Gray, Mathews, and MacRoberts, A Treatise on Bessel Functions (Macmillan and Company, Ltd., London, England, 1931), p. 46.

Jackson, D.

D. Jackson, Fourier Series and Orthogonal Polynomials (The Mathematical Association of America, 1941), pp. 33, 34.

Jenkins,

For example, Jenkins and White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950), second edition, chapter 17.

Kitahara,

Although this progressive disappearance of submaxima has probably been observed in many laboratories, it apparently was not mentioned in the literature until as late as 1938. After the present work was completed our attention was called to an article in the Journal of Science of Hirosima University, Series A,  14, 239–242, Sept.1950 by Takeyama, Kitahara, and Matuhayasi giving a preliminary, qualitative treatment of this problem with a slit source. In this article reference is made to D. Onoyama, Journal of Science of Hirosima University,  8, 81 (1938), who found that the number of secondary maxima diminished, one by one, as the breadth of the slit of the collimator was gradually increased.

MacRoberts,

Gray, Mathews, and MacRoberts, A Treatise on Bessel Functions (Macmillan and Company, Ltd., London, England, 1931), p. 46.

Mathews,

Gray, Mathews, and MacRoberts, A Treatise on Bessel Functions (Macmillan and Company, Ltd., London, England, 1931), p. 46.

Matuhayasi,

Although this progressive disappearance of submaxima has probably been observed in many laboratories, it apparently was not mentioned in the literature until as late as 1938. After the present work was completed our attention was called to an article in the Journal of Science of Hirosima University, Series A,  14, 239–242, Sept.1950 by Takeyama, Kitahara, and Matuhayasi giving a preliminary, qualitative treatment of this problem with a slit source. In this article reference is made to D. Onoyama, Journal of Science of Hirosima University,  8, 81 (1938), who found that the number of secondary maxima diminished, one by one, as the breadth of the slit of the collimator was gradually increased.

Michelson, A. A.

A. A. Michelson and F. G. Pease, Astrophys. J. 53, 249 (1921).
[Crossref]

A. A. Michelson, Phil. Mag. 30, 1 (1890).
[Crossref]

Pease, F. G.

A. A. Michelson and F. G. Pease, Astrophys. J. 53, 249 (1921).
[Crossref]

Takeyama,

Although this progressive disappearance of submaxima has probably been observed in many laboratories, it apparently was not mentioned in the literature until as late as 1938. After the present work was completed our attention was called to an article in the Journal of Science of Hirosima University, Series A,  14, 239–242, Sept.1950 by Takeyama, Kitahara, and Matuhayasi giving a preliminary, qualitative treatment of this problem with a slit source. In this article reference is made to D. Onoyama, Journal of Science of Hirosima University,  8, 81 (1938), who found that the number of secondary maxima diminished, one by one, as the breadth of the slit of the collimator was gradually increased.

White,

For example, Jenkins and White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950), second edition, chapter 17.

Astrophys. J. (1)

A. A. Michelson and F. G. Pease, Astrophys. J. 53, 249 (1921).
[Crossref]

Journal of Science of Hirosima University (1)

Although this progressive disappearance of submaxima has probably been observed in many laboratories, it apparently was not mentioned in the literature until as late as 1938. After the present work was completed our attention was called to an article in the Journal of Science of Hirosima University, Series A,  14, 239–242, Sept.1950 by Takeyama, Kitahara, and Matuhayasi giving a preliminary, qualitative treatment of this problem with a slit source. In this article reference is made to D. Onoyama, Journal of Science of Hirosima University,  8, 81 (1938), who found that the number of secondary maxima diminished, one by one, as the breadth of the slit of the collimator was gradually increased.

Phil. Mag. (1)

A. A. Michelson, Phil. Mag. 30, 1 (1890).
[Crossref]

Other (4)

For example, Jenkins and White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950), second edition, chapter 17.

D. Jackson, Fourier Series and Orthogonal Polynomials (The Mathematical Association of America, 1941), pp. 33, 34.

Gray, Mathews, and MacRoberts, A Treatise on Bessel Functions (Macmillan and Company, Ltd., London, England, 1931), p. 46.

See reference 3, pp. 312–317.

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Figures (6)

Figs. 1–4
Figs. 1–4

The intensity distributions for the cases N=3, 4, 5, 6, respectively, and their dependence upon angular breadth (α) of slit source. The lowest curve in each case is that obtained with a line source. It is drawn to a different scale.

Fig. 5
Fig. 5

The intensity distribution for the case N=4 and its dependence on the angular diameter, 2α, of the disk source. The lowest curve is that for a point source. It is drawn to a different scale.

Fig. 6
Fig. 6

Slit source, N=4. First disappearance of the submaxima is seen in the center photograph.

Fig. 7
Fig. 7

Slit source, N=6. The top photograph is the pattern obtained with a very narrow slit source. In the others the source slit was adjusted to give the characteristic patterns up to the second disappearance of the submaxima.

Fig. 8
Fig. 8

Disk source, N=4. Separation of slits, d, is constant. Diameter of source increased from top to bottom showing N−2, 0, and N−3 submaxima.

Fig. 9
Fig. 9

Disk source, N=4. Source diameter is constant. Slit separation, d, varied showing N−2, 0, and N−3 submaxima.

Equations (15)

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I = R 0 2 sin 2 β β 2 sin 2 N γ sin 2 γ ,
I d i = I 0 sin 2 N γ sin 2 γ ,             γ = [ π d λ ] ( θ + i )
I = - α / 2 + α / 2 I 0 sin 2 N γ sin 2 γ d i .
sin 2 N γ sin 2 γ = 2 [ N 2 + ( N - 1 ) cos 2 γ + ( N - 2 ) cos 4 γ + + cos 2 ( N - 1 ) γ ] .
I = I 0 [ N α + ( N - 1 ) π n cos 2 π n θ sin π n α + + 1 ( N - 1 ) π n cos 2 ( N - 1 ) π n θ sin ( N - 1 ) π n α ] .
I = I 0 π n m = 1 m = N - 1 [ N π n α + ( N - m m ) cos 2 m π n θ sin m π n α ] .
d I = f ( i ) d i sin 2 N γ sin 2 γ ; on expanding this , with n = 2 π d / λ ,
d I = 2 f ( i ) d i { N 2 + ( N - 1 ) [ cos n θ cos n i - sin n θ sin n i ] + + [ cos ( N - 1 ) n θ cos ( N - 1 ) n i - sin ( N - 1 ) n θ sin ( N - 1 ) n i ] } .
f ( i ) d i = 2 K 2 ( α 2 - i 2 ) 1 2 d i .
I = 4 K 2 - α + α { N 2 ( α 2 - i 2 ) 1 2 + ( N - 1 ) ( α 2 - i 2 ) 1 2 × [ cos n θ cos n i - sin n θ sin n i ] + + ( α 2 - i 2 ) 1 2 × [ cos ( N - 1 ) n θ cos ( N - 1 ) n i - sin ( N - 1 ) n θ sin ( N - 1 ) n i ] } d i .
I = 8 K 2 { 0 α N 2 ( α 2 - i 2 ) 1 2 d i + ( N - 1 ) cos n θ × 0 α ( α 2 - i 2 ) 1 2 cos n i d i + + cos ( N - 1 ) n θ × 0 α ( α 2 - i 2 ) 1 2 cos ( N - 1 ) n i d i } .
u = i / α ,             m = n α = 2 π d α / λ ,
I = 8 K 2 α 2 { 0 1 N 2 ( 1 - u 2 ) 1 2 d u + ( N - 1 ) cos n θ × 0 1 ( 1 - u 2 ) 1 2 cos m u d u + + cos ( N - 1 ) n θ × 0 1 ( 1 - u 2 ) 1 2 cos ( N - 1 ) m u d u } ,
I = 8 K 2 α 2 π / 2 { N / 2 + ( N - 1 ) m cos n θ J 1 ( m ) + + 1 ( N - 1 ) m cos ( N - 1 ) n θ J 1 [ ( N - 1 ) m ] } ,
I = 4 K 2 α 2 π m l = 1 l = N - 1 [ N m 4 + ( N - l l ) cos l n θ J 1 ( l m ) ] .