Abstract

An approximate expression for the output beam from a confocal resonator, into which a pinhole for mode selection is inserted at an arbitrary position near the mirror at one end, is studied experimentally. The output beam consists of the fundamental component, in which the diffraction from the pinhole edge is not considered, and a field of the higher order component including the diffraction. In the region of the pinhole aperture larger than the beam width of the confocal resonator before the insertion of the pinhole, the diffraction component can be neglected. The output beam then may be approximated by the beam of the fundamental component, which is expressed simply by the field of the confocal resonator with unequal-sized mirrors of Boyd and Kogelnik. These results are confirmed by both experimental and theoretical examinations.

© 1972 Optical Society of America

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References

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  1. A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963); in Quantum Electronics, P. Grivet, N. Bloembergen, Eds. (Columbia U. P., New York, 1964), Vol. 3, 1263.
    [CrossRef]
  2. T. Miyamoto, K. Yasuura, Tech. Rept. Kyushu Univ. 40, 95 (1967).
  3. T. Miyamoto, K. Yasuura, Appl. Opt. 10, 161 (1971).
    [CrossRef] [PubMed]
  4. G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

1971 (1)

1967 (1)

T. Miyamoto, K. Yasuura, Tech. Rept. Kyushu Univ. 40, 95 (1967).

1963 (1)

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963); in Quantum Electronics, P. Grivet, N. Bloembergen, Eds. (Columbia U. P., New York, 1964), Vol. 3, 1263.
[CrossRef]

1962 (1)

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

Boyd, G. D.

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

Fox, A. G.

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963); in Quantum Electronics, P. Grivet, N. Bloembergen, Eds. (Columbia U. P., New York, 1964), Vol. 3, 1263.
[CrossRef]

Kogelnik, H.

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

Li, T.

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963); in Quantum Electronics, P. Grivet, N. Bloembergen, Eds. (Columbia U. P., New York, 1964), Vol. 3, 1263.
[CrossRef]

Miyamoto, T.

T. Miyamoto, K. Yasuura, Appl. Opt. 10, 161 (1971).
[CrossRef] [PubMed]

T. Miyamoto, K. Yasuura, Tech. Rept. Kyushu Univ. 40, 95 (1967).

Yasuura, K.

T. Miyamoto, K. Yasuura, Appl. Opt. 10, 161 (1971).
[CrossRef] [PubMed]

T. Miyamoto, K. Yasuura, Tech. Rept. Kyushu Univ. 40, 95 (1967).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

G. D. Boyd, H. Kogelnik, Bell Syst. Tech. J. 41, 1347 (1962).

Proc. IEEE (1)

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963); in Quantum Electronics, P. Grivet, N. Bloembergen, Eds. (Columbia U. P., New York, 1964), Vol. 3, 1263.
[CrossRef]

Tech. Rept. Kyushu Univ. (1)

T. Miyamoto, K. Yasuura, Tech. Rept. Kyushu Univ. 40, 95 (1967).

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Figures (9)

Fig. 1
Fig. 1

Output beams from a confocal resonator including a pinhole. Resonator spacing, M 1 M 2 ¯ = b; position of pinhole, M 2 P ¯ = l.

Fig. 2
Fig. 2

Measurement apparatus for beam parameters.

Fig. 3
Fig. 3

Photographs of the output beams from a confocal resonator including a pinhole and measurement of the radius of curvature of the wavefront (b = 150 cm, l = 10 cm, a2 = 0.5 mm, and ws2′ = 0.51 mm). Measured values of R are obtained less about 10 cm from the value in the case where the ideal mirror is used, because in this experiment we have used a mirror whose back surface has been made flat and the output beam has been enlarged by its property of concave lens.3

Fig. 4
Fig. 4

Photographs of the output beams from a confocal resonator when the pinhole size is varied, and measurement of the radius of curvature of the wavefront in the output beam (b = 100 cm; l = 2 cm; a2 = 0.50 mm, 0.65 mm, 0.75 mm; ws2′ = 0.44 mm).

Fig. 5
Fig. 5

Photographs of the output beams and measurement of the radius of curvature of the wavefront of the diffraction component in the output beam from (a) M2 and (b) M1.

Fig. 6
Fig. 6

Beams in a confocal resonator when a pinhole is inserted (a) close to M2 and (b) at an arbitrary point. — - —, dominant mode before insertion of the pinhole; —, fundamental component; - - - -, higher order component.

Fig. 7
Fig. 7

Normalized amplitude distributions of each beam when the pinhole size is varied (b = 150 cm; l = 10 cm; a1 = 1.36 mm; a2 = 0.3 mm, 0.5 mm, 0.6 mm, and 0.7 mm; ws2′ = 0.514 mm).

Fig. 8
Fig. 8

Confocal resonator system.

Fig. 9
Fig. 9

Normalized amplitude distributions of the dominant mode in a confocal resonator with small Fresnel number. —, Calculated by iterative method; – – –, Gaussian distribution in the same Fresnel number. Each curve shows Fresnel number of (a) 1.0/2π, (b) 2.25/2π, (c) 4.0/2π, (d) 6.76/2π, and (e) 17.64/2π.

Tables (1)

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Table I Comparison of Beam Parameters of Experimental and Theoretical Data

Equations (9)

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R = ( w n + 1 2 - w n 2 ) / 2 λ ,
U ( z , w ) exp [ - w 2 / w s ( z ) ] · exp { - j k [ z + w 2 / 2 R ( z ) + C ( z ) ] } ,
w s ( z ) = { b 0 [ 1 + ( 2 z / b 0 ) 2 ] / k } 1 2 ,             R ( z ) = z + b 0 2 / 4 z ,
b 0 2 = 2 b d 1 - d 1 2 ,             d 1 = b - 2 D ,             z = z - D ,
D = ( b / 2 ) [ 1 - ( a 3 / a 1 ) 2 ] / [ 1 + ( a 3 + a 1 ) 2 ] ,
w s 4 2 = b 0 [ 1 + ( d 1 / b 0 ) 2 ] / k = K a 4 2 , w s 1 2 = b 0 [ 1 + ( d 1 - 2 l ) 2 / b 0 2 ] / k = K a 1 2 , w s 2 2 = b 0 [ 1 + ( d 2 - 2 l ) 2 / b 0 2 ] / k = K a 2 2 , w s 3 2 = b 0 [ 1 + ( d 2 / b 0 ) 2 ] / k = K a 3 2 ,
[ a 4 2 a 3 2 ] = 1 A 2 A 2 - B 2 B 2 [ A 2 , - B 2 - B 2 , A 2 ] · [ a 1 2 a 2 2 ] .
a 4 2 a 1 2 / A 2 - a 2 2 ( B 2 / A 2 A 2 ) .
a 3 2 ( a 2 2 - a 1 2 B 2 ) / ( 1 - B ) 2 .

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