Abstract

A gradient-index rod whose length is a quarter of the periodic length works as a retroreflective element when its endface is coated with metal. This paper describes the experiment and theory on the reflectivity of such a rod and that of an array comprising seven or nineteen rods when they are illuminated by a Gaussian laser beam. We clarify how the reflectivity depends on the offset of the beam axis and the inclination angle of the axis. The far field pattern of the reflected wave is also investigated.

© 1991 Optical Society of America

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References

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  1. K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Microoptics (Academic, New York, 1984).
  2. M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), p. 289.
  3. S. Nemoto, “Determination of Waist Parameters of a Gaussian Beam,” Appl. Opt. 25, 3859–3863 (1986).
    [CrossRef] [PubMed]
  4. S. M. Watson, J. P. Mills, S. K. Rogers, “Sidelobe Reduction via Multiaperture Optical Systems,” Appl. Opt. 28, 687–693 (1989).
    [CrossRef] [PubMed]
  5. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 22.

1989

1986

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), p. 289.

Iga, K.

K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Microoptics (Academic, New York, 1984).

Kokubun, Y.

K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Microoptics (Academic, New York, 1984).

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 22.

Mills, J. P.

Nemoto, S.

Oikawa, M.

K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Microoptics (Academic, New York, 1984).

Rogers, S. K.

Watson, S. M.

Appl. Opt.

Other

K. Iga, Y. Kokubun, M. Oikawa, Fundamentals of Microoptics (Academic, New York, 1984).

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), p. 289.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 22.

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Figures (20)

Fig. 1
Fig. 1

Retroreflective property of a gradient-index rod whose length is a quarter of the periodic length Lp and whose endface is coated with metal.

Fig. 2
Fig. 2

Configurations for measuring the reflectivities of the rod and rod arrays.

Fig. 3
Fig. 3

Cross sections of the rod and rod arrays (solid circles) along with the 1/e2 intensity spots of the incident beams (broken circles).

Fig. 4
Fig. 4

(a) Reflectivity of the rod as a function of the beam offset x0 for θ0 = 0. Open circles indicate the measured values, and the curve R1 represents the theoretical result. (b) Difference between the experimental and theoretical results shown in (a). The curve Ls represents the scattering loss due to the rod edge.

Fig. 5
Fig. 5

Cross section of the rod and the 1/e4 intensity spot of the incident beam for calculating the scattering loss due to the rod edge.

Fig. 6
Fig. 6

Reflectivity of the rod as a function of the inclination angle θ0 for x0 = 0. Open circles indicate the measured values, and the curve R1 represents the theoretical result.

Fig. 7
Fig. 7

Arrays comprising (a) seven and (b) nineteen rods.

Fig. 8
Fig. 8

Reflectivity of the seven-rod array as a function of x0 for d = 300 cm and θ0 = 0. Two curves R7 represent the theoretical results for the array displaced to the horizontal h and vertical υ directions with respect to the incident beam.

Fig. 9
Fig. 9

Reflectivities of the seven-rod and nineteen-rod arrays as functions of θ0 for x0 = 0. Open and solid circles indicate the measured values, and the two curves represent the theoretical results.

Fig. 10
Fig. 10

Reflectivity of the nineteen-rod array as a function of x0 for d = 900 cm and θ0 = 0. The array was displaced in the horizontal direction.

Fig. 11
Fig. 11

Reflectivities of the rod, seven-rod array, and nineteen-rod array as functions of θ0 obtained when the spot size of the incident beam equals the rod radius or array radius.

Fig. 12
Fig. 12

Far field patterns of the wave reflected by a single rod observed when the spot size of the beam is nearly one-half of the rod radius and (a) x0 = 0, θ0 = 0, (b) x0 = 100 μm, θ0 = 0, (c) x0 = 0, θ0 = 5°; (d) intensity profile of the pattern (a).

Fig. 13
Fig. 13

(a) Michelson interferometer for observing the interference fringe due to the waves reflected by the mirror or rod. The fringes were observed when (b) two flat mirrors, (c) one flat mirror and one rod, and (d) two rods were used on the two arms of the interferometer.

Fig. 14
Fig. 14

(a) Far field pattern of the wave reflected by the seven-rod array; (b) intensity profile of the pattern.

Fig. 15
Fig. 15

(a) View of the corridor where the nineteen-rod array is placed 25 m away from the laser head; (b) close-up of the array shining with the laser beam.

Fig. 16
Fig. 16

(a) Far field pattern of the wave reflected by the nineteen-rod array placed 25 m away from the laser head. This pattern was observed on a white sheet of paper placed adjacent to the laser head. (b) Intensity profile of the pattern observed using the configuration of Fig. 2(b).

Fig. 17
Fig. 17

Reflectivity of a corner cube as a function of the beam offset x0 for θ0 = 0. Two states of the cube orientation are depicted. Broken circles indicate the 1/e2 intensity spots of the incident beams.

Fig. 18
Fig. 18

Reflectivity of a corner cube as a function of the inclination angle θ0 for x0 = 0.

Fig. 19
Fig. 19

Far field pattern of the wave reflected by a corner cube whose orientation is depicted at the right in Fig. 18. The pattern was observed when x0 = 0 and θ0 = 0.

Fig. 20
Fig. 20

(a) Interface between the air and rod. (b) Effective radius ae for an inclined aperture. The area of an inclined aperture at the left equals that of a circular aperture at the right.

Equations (13)

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n ( r ) = n 0 [ 1 ( g r ) 2 ] 1 / 2 , 0 r a ,
R = ( 1 / T H R H T L ) ( P 3 / P 0 ) ,
L s = A C exp [ 2 ( ρ / w ) 2 ] d s ,
ρ = ( a 2 2 a x 0 cos ϕ + x 0 2 ) 1 / 2 ,
ϕ 1 = tan 1 [ ( a 2 x 1 2 ) 1 / 2 / x 1 ] ,
x 1 = ( x 0 2 + a 2 2 w 2 ) / 2 x 0 ,
a 2 w < x 0 < a + 2 w .
R = ( 1 / R M 1 R M 2 T H R H T L ) ( P 3 / P 0 ) ,
T = 4 n 1 cos θ i ( n 2 2 n 1 2 sin 2 θ i ) 1 / 2 × [ n 1 cos θ i + ( n 2 2 n 1 2 sin 2 θ i ) 1 / 2 ] 2 .
T 1 = 4 cos θ 0 ( n 0 2 sin 2 θ 0 ) 1 / 2 [ cos θ 0 + ( n 0 2 sin 2 θ 0 ) 1 / 2 ] 2 .
T 2 = 4 n 0 cos θ 1 ( 1 n 0 2 sin 2 θ 1 ) 1 / 2 × [ n 0 cos θ 1 + ( 1 n 0 2 sin 2 θ 1 ) 1 / 2 ] 2 .
n 0 sin θ 1 = sin θ 0 , n 0 cos θ 1 = ( n 0 2 sin 2 θ 0 ) 1 / 2 .
R d = R m ( 4 cos θ 0 ) 2 ( n 0 2 sin 2 θ 0 ) [ cos θ 0 + ( n 0 2 sin 2 θ 0 ) 1 / 2 ] 4 .

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