Abstract

A simple method for measuring the beam parameters of a laser beam, using a hologram, is proposed, and the accuracy is examined. By measuring the beam width and radius of curvature of the wavefront of a He–Ne gas laser, including several higher order modes, the behavior of a laser beam is investigated experimentally. As another application of this method, the phase distribution of the diffracted field from an aperture and a slit illuminated by a laser beam is measured, and the effects of a beam upon the diffracted field are studied experimentally.

© 1971 Optical Society of America

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References

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  1. J. H. Harris, M. P. Givens, J. Opt. Soc. Amer. 59, 862 (1966).
  2. J. T. Ruscio, Bell Syst. Tech. J. 45, 1583 (1966).
  3. T. Miyamoto, K. Yasuura, Tech. Rep. Kyushu U. 40, 95 (1967).
  4. G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
  5. T. Miyamoto, K. Yasuura, Tech. Rep. Kyushu U. 41, 692 (1968).
  6. F. A. Abramskii, Opt. Spectrosc. 22, 333 (1967).
  7. D. C. W. Morley, L. Allen, D. C. G. Jones, Phys. Lett. 19, 484 (1965).
    [CrossRef]

1968 (1)

T. Miyamoto, K. Yasuura, Tech. Rep. Kyushu U. 41, 692 (1968).

1967 (2)

F. A. Abramskii, Opt. Spectrosc. 22, 333 (1967).

T. Miyamoto, K. Yasuura, Tech. Rep. Kyushu U. 40, 95 (1967).

1966 (2)

J. H. Harris, M. P. Givens, J. Opt. Soc. Amer. 59, 862 (1966).

J. T. Ruscio, Bell Syst. Tech. J. 45, 1583 (1966).

1965 (1)

D. C. W. Morley, L. Allen, D. C. G. Jones, Phys. Lett. 19, 484 (1965).
[CrossRef]

1961 (1)

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

Abramskii, F. A.

F. A. Abramskii, Opt. Spectrosc. 22, 333 (1967).

Allen, L.

D. C. W. Morley, L. Allen, D. C. G. Jones, Phys. Lett. 19, 484 (1965).
[CrossRef]

Boyd, G. D.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

Givens, M. P.

J. H. Harris, M. P. Givens, J. Opt. Soc. Amer. 59, 862 (1966).

Gordon, J. P.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

Harris, J. H.

J. H. Harris, M. P. Givens, J. Opt. Soc. Amer. 59, 862 (1966).

Jones, D. C. G.

D. C. W. Morley, L. Allen, D. C. G. Jones, Phys. Lett. 19, 484 (1965).
[CrossRef]

Miyamoto, T.

T. Miyamoto, K. Yasuura, Tech. Rep. Kyushu U. 41, 692 (1968).

T. Miyamoto, K. Yasuura, Tech. Rep. Kyushu U. 40, 95 (1967).

Morley, D. C. W.

D. C. W. Morley, L. Allen, D. C. G. Jones, Phys. Lett. 19, 484 (1965).
[CrossRef]

Ruscio, J. T.

J. T. Ruscio, Bell Syst. Tech. J. 45, 1583 (1966).

Yasuura, K.

T. Miyamoto, K. Yasuura, Tech. Rep. Kyushu U. 41, 692 (1968).

T. Miyamoto, K. Yasuura, Tech. Rep. Kyushu U. 40, 95 (1967).

Bell Syst. Tech. J. (2)

J. T. Ruscio, Bell Syst. Tech. J. 45, 1583 (1966).

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

J. Opt. Soc. Amer. (1)

J. H. Harris, M. P. Givens, J. Opt. Soc. Amer. 59, 862 (1966).

Opt. Spectrosc. (1)

F. A. Abramskii, Opt. Spectrosc. 22, 333 (1967).

Phys. Lett. (1)

D. C. W. Morley, L. Allen, D. C. G. Jones, Phys. Lett. 19, 484 (1965).
[CrossRef]

Tech. Rep. Kyushu U. (2)

T. Miyamoto, K. Yasuura, Tech. Rep. Kyushu U. 41, 692 (1968).

T. Miyamoto, K. Yasuura, Tech. Rep. Kyushu U. 40, 95 (1967).

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Figures (11)

Fig. 1
Fig. 1

Measurement apparatus.

Fig. 2
Fig. 2

Behavior of the measured value of radius of curvature of the wavefront, when the beam enlarging system is misadjusted by ΔL. Solid line indicates calculated value. (For the relation between M and ΔL, see Appendix.)

Fig. 3
Fig. 3

Radius of curvature of the wavefront in the rear of the focus of a convex lens (f = 20 cm). Dots indicate measured values.

Fig. 4
Fig. 4

Hologram in each mode and microphotometer output.

Fig. 5
Fig. 5

Radii of curvature of the wavefronts of laser beams from two kinds of typical resonator mirrors. The value of the refractive index of the mirror used in this paper is 1.514.

Fig. 6
Fig. 6

Measurement of the amplitude distribution and beam width in the TEM00 mode.

Fig. 7
Fig. 7

Radius of curvature of the wavefront in several higher order modes.

Fig. 8
Fig. 8

Diffraction field from a slit of width 2a in the case of a/we ≪ 1. (A) Holograms, (B) Radius of curvature of the wavefront. In (A), (a), (b), and (c) indicate the cases where the distances from the aperture to the observing points are 73 cm, 108 cm, and 150 cm, respectively, and the case of 2a = 0.05 mm.

Fig. 9
Fig. 9

Diffraction field from a circular aperture of diameter 2a in the case of a/we ≪ 1. (A) Diffraction patterns and their holograms, (B) Wavefront (phase distribution), (C) Radius of curvature of the wavefront. (a)–(c) show the cases where the distances from the aperture to the observing points are 96 cm, 146 cm, and 200 cm, respectively. The number indicates the position of each bright ring in the diffraction pattern, counting from the center. Arrows indicate the calculated positions of w0 which satisfy J1(kaw0/z) = 0.

Fig. 10
Fig. 10

Phase distribution of the diffraction field by several circular apertures [(a) 1 mmΦ, (b) 2 mmΦ, (c) 3 mmΦ, (d) 4 mmΦ (e) no aperture] illuminated by laser beam of we = 2 mm and by an approximate plane wave.

Fig. 11
Fig. 11

Estimation of M(z) in the beam enlarging system.

Tables (1)

Tables Icon

Table I Measured and Calculated Values of the Beam Parameters and the Angle of Beam Divergence of a Laser Beam in Each Mode

Equations (14)

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E m n ( x , y , z ) H m [ 2 1 2 x / w e ( z ) ] H n [ 2 1 2 y / w e ( z ) ] × exp [ - ( x 2 + y 2 ) / w e 2 ( z ) ] exp [ - j φ ( x , y , z ) ] ,
φ ( x , y , z ) = k z + k ( x 2 + y 2 ) / 2 R ( z ) + k b / 2 + ( 1 + m + n ) × { tan - 1 [ ( b - 2 z ) / ( b + 2 z ) ] - ( π / 2 ) } ,
w e ( z ) = [ b ( 1 + 4 z 2 / b 2 ) / k ] 1 2 , R ( z ) = z + b 2 / 4 z ,             b 2 = 2 b d - d 2 ,
w e ( z ) = [ b ( 1 + 4 z 2 / b 2 ) / k ] 1 2 , R ( z ) = z + b 2 / 4 z ,             b 2 = 2 b d / N - d 2 ,
d = 2 N b d / [ 2 b + d ( N 2 - 1 ) ] ,             z = z - ( d - d ) / 2 ,
I ( z , w ) = β 2 + α 2 { exp [ - 2 w 2 / w e 2 ( z ) ] + 2 ρ exp [ - w 2 / w e 2 ( z ) ] × cos [ φ s ( z , w ) - φ r ( z , w ) ] } ,
φ s ( z , w ) = k z + [ k w 2 / 2 R s ( z ) ] + c s , φ r ( z , w ) = k z + [ k w 2 / 2 R r ( z ) ] + c r , w 2 = x 2 + y 2 ,             α , β , c s , c r = constants ,             ρ = α / β ,
φ s ( z , w q ) - φ r ( z , w q ) = k w q 2 [ 1 - 1 / M ( z ) ] / 2 R s ( z ) + k γ = 2 q π ,
R s ( z ) = ( w q + 1 2 - w q 2 ) / 2 λ .
arg [ E m o ( x , z ) ] = φ ( x , z ) + { 0 , H m ( X ) > 0 , π , H m ( X ) < 0 , X = 2 1 2 x / w e ( z ) ,
R s ( z ) = R s ( z ) / [ 1 - M ( z ) ] ;
U ( ξ ) = C exp ( j k s ) sin [ k a ( ξ / z ) ] / k a ( ξ / z ) , arg [ U ( ξ ) ] = k s + { 0 , sin [ k a ( ξ / z ) ] > 0 , π , sin [ k a ( ξ / z ) ] < 0.
M = R s / R τ - f 1 2 / f 2 2 + [ R s ( 1 - f 1 / R s ) / f 2 2 ] Δ L ,
Δ l / D l / R r ,             M = R s / R τ ( R s / D ) ( Δ l / l ) .

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