Abstract

We used the focal line of a cylindrical lens illuminated by a collimated expanded He–Ne red laser light as a filamentary light source to observe Young’s interference fringes. When the monochromatic filamentary light source is parallel to Young’s slits, the interference fringes are parallel to them and are well defined and highly contrasted. When the filamentary light source is rotated in a plane parallel to that of the slits, the visibility of the interference fringes is always extremely high, but their geometrical distribution and the distance between them depend on the rotation angle and the observation distance. A simple theoretical model is proposed, based on a cylindrical wave propagated according to the Huygens–Kirchhoff formula. The experimental results coincide remarkably well with the stated theoretical model.

© 1999 Optical Society of America

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References

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  1. M. Born, E. Wolfe, Principles of Optics, 5th ed. ( Pergamon, Oxford, 1975).

Born, M.

M. Born, E. Wolfe, Principles of Optics, 5th ed. ( Pergamon, Oxford, 1975).

Wolfe, E.

M. Born, E. Wolfe, Principles of Optics, 5th ed. ( Pergamon, Oxford, 1975).

Other (1)

M. Born, E. Wolfe, Principles of Optics, 5th ed. ( Pergamon, Oxford, 1975).

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Figures (7)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

Definition of coordinate systems.

Fig. 3
Fig. 3

Cylindrical wave front arriving at the slits’ plane.

Fig. 4
Fig. 4

Coordinate systems and fringes in the observation plane.

Fig. 5
Fig. 5

Young’s fringes for a monochromatic coherent filamentary light source (focal line of a collimated laser beam); photographs were taken with a lensless camera at different distances from the slits’ plane with different rotations of the slits. Note that the narrow fringes seen within Young’s fringes correspond to interference of the wave reflected once into the cylindrical lens with the one transmitted directly.

Fig. 6
Fig. 6

Angle ψ of rotation of interference fringes as a function of ϕ (angle of rotation of the slits) and R (distance from the slits’ plane to the observation plane). Experimental points were measured in Fig. 5, and the theoretical expression is from Eq. (7).

Fig. 7
Fig. 7

Distance d between interference fringes as a function of angle ϕ between the slits and the source for several distances R between the slits’ plane and the observation plane. Experimental points were measured in Fig. 5, and the theoretical expression is from Eq. (12).

Equations (13)

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A(xs, ys)=exp[iπ(xs sin ϕ+ys cos ϕ)2]ΛL.
A(xo, yo)=slits exp[iδ(xs, ys)]exp(i2πρ/λ)ρ dxsdys,
ρ=R{1+12[(xo-xs)/R]2+12[(yo-ys)/R]2}
A(xo, yo)=slitsdxsdys exp iπ(xs sin ϕ+ys cos ϕ)2λL+(xo-xs)2+(yo-ys)2λR.
A(xo, yo)=1L/R+sin2 ϕ ×cos πaλR (sin ϕ cos ϕ)L/R+sin2 ϕ xo-yo.
yo=sin ϕ cos ϕL/R+sin2 ϕ xo+n 2λRa
yo=(1+R/L)tan ϕxo+n 2λRa (1+R/L sin2 ϕ)cos ϕ
tan ψ=(1+R/L)tan ϕ,
yo=yo=2nλR/a;
yo=2λRa [1+(R/L)sin2 ϕ]cos ϕ,
yo=2λRa,
d=2λRa cos(ψ-ϕ).
d=2λRa cos{arctan n[(1+R/L)tan ϕ]-ϕ}.

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