Abstract

The propagation characteristics of a beam diffracted by a circular aperture are investigated. The beam-quality factor M 2 defined by an 86.5% power-content radius is given theoretically and experimentally as a function of the truncation ratio. It is found that the theoretical limit of M 2 is 2.37 times as great as that of an incident beam as the truncation ratio approaches 0. For a weakly diffracted beam a simple formula giving M 2 is derived. Although M 2 does not increase much with diffraction, the influence of diffraction should be taken into account in beam brightness.

© 2002 Optical Society of America

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References

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  1. P. Belland, J. P. Crenn, “Changes in the characteristics of a Gaussian beam weakly diffracted by a circular aperture,” Appl. Opt. 21, 522–527 (1982).
    [CrossRef] [PubMed]
  2. E. M. Drege, N. G. Skinner, D. M. Byrne, “Analytical far-field divergence angle of a truncated Gaussian beam,” Appl. Opt. 39, 4918–4925 (2000).
    [CrossRef]
  3. K. Tanaka, N. Saga, K. Hauchi, “Focusing of a Gaussian beam through a finite aperture lens,” Appl. Opt. 8, 1098–1101 (1985).
    [CrossRef]
  4. Y. Li, “Oscillations and discontinuity in the focal shift of Gaussian laser beams,” J. Opt. Soc. Am. A 3, 1761–1765 (1986).
    [CrossRef]
  5. R. G. Wenzel, “Effect of the aperture-lens separation on the focal shift in large-F-number systems,” J. Opt. Soc. Am. A 4, 340–345 (1987).
    [CrossRef]
  6. A. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
  7. “Lasers and laser-related equipment—test methods for laser beam parameters—beam widths, divergence angle, and beam propagation factor,” ISO 11146: 1999(E) (International Organization for Standardization, Geneva, Switzerland, 1999).
  8. H. T. Yura, T. S. Rose, “Gaussian beam transfer through hard-aperture optics,” Appl. Opt. 34, 6826–6828 (1995).
    [CrossRef] [PubMed]

2000

1995

1987

1986

1985

K. Tanaka, N. Saga, K. Hauchi, “Focusing of a Gaussian beam through a finite aperture lens,” Appl. Opt. 8, 1098–1101 (1985).
[CrossRef]

1982

Belland, P.

Byrne, D. M.

Crenn, J. P.

Drege, E. M.

Hauchi, K.

K. Tanaka, N. Saga, K. Hauchi, “Focusing of a Gaussian beam through a finite aperture lens,” Appl. Opt. 8, 1098–1101 (1985).
[CrossRef]

Li, Y.

Rose, T. S.

Saga, N.

K. Tanaka, N. Saga, K. Hauchi, “Focusing of a Gaussian beam through a finite aperture lens,” Appl. Opt. 8, 1098–1101 (1985).
[CrossRef]

Siegman, A.

A. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).

Skinner, N. G.

Tanaka, K.

K. Tanaka, N. Saga, K. Hauchi, “Focusing of a Gaussian beam through a finite aperture lens,” Appl. Opt. 8, 1098–1101 (1985).
[CrossRef]

Wenzel, R. G.

Yura, H. T.

Appl. Opt.

J. Opt. Soc. Am. A

Other

A. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).

“Lasers and laser-related equipment—test methods for laser beam parameters—beam widths, divergence angle, and beam propagation factor,” ISO 11146: 1999(E) (International Organization for Standardization, Geneva, Switzerland, 1999).

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Figures (8)

Fig. 1
Fig. 1

Schematic illustration of the setup for measuring a laser-beam diameter. The conditions are as follows: z 1 = 754 mm, z 0 = 720 mm (z 0f), 600 mm (z 0 = f), and f = 600 mm. From the measured results, w 0 = 0.29 mm, w = 0.6 mm, and M 2 = 1 of the He-Ne laser.

Fig. 2
Fig. 2

Propagations of the diffracted beams in the case of (a) z 0f and (b) z 0 = f. The truncation ratio is as follows: ●, infinite; ○, 1.5; □, 1; ◇, 0.4. Infinite means no aperture.

Fig. 3
Fig. 3

Focal shift of the diffracted beam: ●, ■, measured data at z 0f and z 0 = f, respectively; dotted curves, calculation results obtained from Eq. (14).

Fig. 4
Fig. 4

Spot diameter of the diffracted beam at z 0 = f: ●, measured data; ○, data corrected by F; solid curve, calculation results of Eq. (25); dotted curve, results of numerical integration.

Fig. 5
Fig. 5

Percentage error of the spot diameter calculated with Eq. (25) with reference to the numerical integration.

Fig. 6
Fig. 6

Full-beam divergence of the diffracted beam: ●, measured data; solid curve, calculation results from Eq. (29).

Fig. 7
Fig. 7

Beam-quality factor of a diffracted beam normalized by that of the incident beam: ●, ■, measured data at z 0f and z 0 = f, respectively; ○, □, data corrected by F; solid curve, calculation results of Eq. (30); dotted curve, results of numerical integration.

Fig. 8
Fig. 8

Diffracted beam brightness (right) and transmitted power (left) normalized by the incident beam brightness and power, respectively. The brightness is calculated from Eq. (48), and its results are valuable in the region of (a/ω) > 0.7.

Equations (56)

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ω=ω01+z1zR21/2.
R=z1+zR2z1,
zR=πω02λ.
λ=M2λ0,
Ur1, z1=exp-r12ω2+ikr122R,
Ur0, z0=2π expikz0iλz00adr1r1Ur1, z1×expik2z0r02+r12J0kr0r1z0,
Ur0, z0=Ur0, z0exp-ikr022f,
Ur, z=2π expikziλz0dr0r0Ur0, z0×expik2zr02+r2J0kr0rz.
Uz=2π expikziλz0dr0r0Ur0, z0expikr022z.
Uz=Gz1+i 1πFza2ω2expikz0+z×expiπFz-a2ω2-1,
Fz=a2λ1z0-zfz-f+1R,
Gz=RfR+z0z-f-zf.
Iz=|Uz|2=Gz21+1πFz2a2ω22exp-2 a2ω2-2 cosπFzexp-a2ω2+1.
2 πFzπFz2+a2ω22-1πFzGzz0f-1×cosha2ω2-cosπFz=sinπFz.
2 πFzπFz2+a2ω22cosha2ω2-cosπFz=sinπFz.
z=fR+z0R+z0-f.
P=πd22I.
P=πω21-u,
u=exp-2 a2ω2.
d=2ωIz1/21-u1/2.
q=z-fR+z0R+z0-f.
Fq=-a2λf2 q,
Gq=fq,
Iq=f2q2+λf2πω22u-2ucosa2πλf2 q+1.
ω=dq=02=1+u1-u1/2λfπω.
0rexp-2 r02ω22πr0dr00aexp-2 r02ω22πr0dr0=0.865,
r=ω12ln11-0.8651-u1/2.
θ=2rf,
θ=2ωf12ln11-0.8651-u1/2.
M2=π4λ0 θd =121+u1-uln11-0.8651-u1/2M2.
dz2=Az2+Bz+C,
zspot=-B2A,
d=C-B24A1/2,
θ=A,
M2=AC-B241/2.
0ω |Uz, r|2dr0 |Uz, r|2dr=0.865.
F=1-0.162 exp-0.25aω-0.164aω2-0.873aω3.
P=π25.09×d22I.
P=πa2×1.
d=5.09 aI.
Iq=sin12a2πλf2 q12a2πλf2 q2a2πλf22
Iq=0=a2πλf22.
d=5.09 λfπa.
θ=0.8651/22af=1.86 af.
M2=2.37M2.
B=P0λ0M22.
P=P01-u.
B=Pλ0M22=2 1-u2ln11-0.8651-u B.
Iz=I0πa2λz22JπXπX2,
πX=πaλθ0.
πX=5.09.
θ0=5.09 λaπ.
d022πa2π=0.865
d0=1.86a.
θ0=df,
θ=d0f.

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