Abstract

The relations for position, spot size, and inclination of the major axis of an elliptical Gaussian beam to knife-edge scanning data are derived. A knife-edge whose scanning direction is adjustable to any angle has been employed to scan across a beam in at least three directions. Nonlinear least-squares fit programs have been developed to check whether a beam is Gaussian, and to evaluate the parameters, with errors, of such an elliptic spot. The evolution of an astigmatic beam in the tangential and sagittal plane is measured.

© 1985 Optical Society of America

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References

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  1. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Claviere, E. A. Franke, J. M. Franke, “Technique for Fast Measurement of Gaussian Laser Beam Parameters,” Appl. Opt. 10, 2775 (1971);Y. Suzaki, A. Tachibana, “Measurement of the μm Sized Radius of Gaussian Laser Beam Using the Scanning Knife-Edge,” Appl. Opt. 14, 2809 (1975);J. M. Khosrofian, B. A. Garetz, “Measurement of a Gaussian Laser Beam Diameter Through the Direct Inversion of Knife-Edge Data,” Appl. Opt. 22, 3406 (1983);M. Mauck, “Knife-Edge Profiling of Q-Switched Nd:YAG Laser Beam and Waist,” Appl. Opt. 18, 599 (1979).
    [CrossRef] [PubMed]
  2. The required size of the detector and its distance from the edge may be estimated as follows: An aperture of a diameter 4 w passes over 99.9% of a circular Gaussian beam of spot size w. As far as diffraction effects due to the edge are concerned, the power of the cylindrical waves in the shadow region diminishes as (kz)−1 [M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 11], which amounts to ∼10−5 for λ = 633 nm at r = 1 cm off the edge: with a detector placed more than 1 cm behind the edge we can safely disregard diffraction effects.
  3. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 7.
  4. Ref. 3, Chap. 26. The approximation 26.2.16 therein is over 2 orders of magnitude more accurate than that used by Khosrofian and Garetz (see Ref. 1); however the following approximation, which is similar to the latter in structure is also accurate to ±1 × 10−5 for −∞⩽x⩽+∞:Q(x)≃{1+exp[f(x)]}−1,f(x)=a1x+a2x3+a3x5+a4x7,a1=1.595700,a2=0.072953,a3=−0.000324,a4=−0.0000350.
  5. Error function paper is available as graph paper no. 46800 from Keuffel & Esser, covering relative magnitudes from 0.01 to 99.99; a better-suited variant of it, covering 0.15–0.85, was devised by D. Preonas, Dayton Research Institute, in 1980. This latter graph paper makes use of the central portion of the Gaussian only, thus avoiding errors due to deviations of the beam from Gaussian shape in the tails.
  6. W. C. Hamilton, Statistics in Physical Science (Ronald Press, New York, 1964), Chaps. 4 and 5.
  7. The program is available on request.
  8. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976), Sec.9.9.
  9. Ref. 8, p. 169, Eq. (9q).
  10. This situation prevails in eigenmodes of a ring laser.
  11. T. D. Baxter, T. T. Saito, G. L. Shaw, R. T. Evans, R. A. Motes, “Mode Matching for a Passive Resonant Ring Laser Gyroscope,” Appl. Opt. 22, 2487 (1983).
    [CrossRef] [PubMed]

1983 (1)

1971 (1)

Arnaud, J. A.

Baxter, T. D.

Born, M.

The required size of the detector and its distance from the edge may be estimated as follows: An aperture of a diameter 4 w passes over 99.9% of a circular Gaussian beam of spot size w. As far as diffraction effects due to the edge are concerned, the power of the cylindrical waves in the shadow region diminishes as (kz)−1 [M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 11], which amounts to ∼10−5 for λ = 633 nm at r = 1 cm off the edge: with a detector placed more than 1 cm behind the edge we can safely disregard diffraction effects.

de la Claviere, B.

Evans, R. T.

Franke, E. A.

Franke, J. M.

Hamilton, W. C.

W. C. Hamilton, Statistics in Physical Science (Ronald Press, New York, 1964), Chaps. 4 and 5.

Hubbard, W. M.

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976), Sec.9.9.

Mandeville, G. D.

Motes, R. A.

Saito, T. T.

Shaw, G. L.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976), Sec.9.9.

Wolf, E.

The required size of the detector and its distance from the edge may be estimated as follows: An aperture of a diameter 4 w passes over 99.9% of a circular Gaussian beam of spot size w. As far as diffraction effects due to the edge are concerned, the power of the cylindrical waves in the shadow region diminishes as (kz)−1 [M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 11], which amounts to ∼10−5 for λ = 633 nm at r = 1 cm off the edge: with a detector placed more than 1 cm behind the edge we can safely disregard diffraction effects.

Appl. Opt. (2)

Other (9)

The required size of the detector and its distance from the edge may be estimated as follows: An aperture of a diameter 4 w passes over 99.9% of a circular Gaussian beam of spot size w. As far as diffraction effects due to the edge are concerned, the power of the cylindrical waves in the shadow region diminishes as (kz)−1 [M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 11], which amounts to ∼10−5 for λ = 633 nm at r = 1 cm off the edge: with a detector placed more than 1 cm behind the edge we can safely disregard diffraction effects.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 7.

Ref. 3, Chap. 26. The approximation 26.2.16 therein is over 2 orders of magnitude more accurate than that used by Khosrofian and Garetz (see Ref. 1); however the following approximation, which is similar to the latter in structure is also accurate to ±1 × 10−5 for −∞⩽x⩽+∞:Q(x)≃{1+exp[f(x)]}−1,f(x)=a1x+a2x3+a3x5+a4x7,a1=1.595700,a2=0.072953,a3=−0.000324,a4=−0.0000350.

Error function paper is available as graph paper no. 46800 from Keuffel & Esser, covering relative magnitudes from 0.01 to 99.99; a better-suited variant of it, covering 0.15–0.85, was devised by D. Preonas, Dayton Research Institute, in 1980. This latter graph paper makes use of the central portion of the Gaussian only, thus avoiding errors due to deviations of the beam from Gaussian shape in the tails.

W. C. Hamilton, Statistics in Physical Science (Ronald Press, New York, 1964), Chaps. 4 and 5.

The program is available on request.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976), Sec.9.9.

Ref. 8, p. 169, Eq. (9q).

This situation prevails in eigenmodes of a ring laser.

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Figures (9)

Fig. 1
Fig. 1

Elliptical beam spot centered at x0,y0 with axes wa and wb tilted against the positive x axis (horizontal) by α0. The beam goes into the paper plane (positive z axis). The scanning edge is placed at xs.

Fig. 2
Fig. 2

Relative power P(xs)/P0 vs scanner position xs for two spot sizes w1 (= 1.5 units on the abscissa) and w2 (= 3 units on the abscissa). The shapes of these curves are identical, except for a difference in w, for scanning of an arbitrarily placed elliptical beam. The beam centers are assumed to be at xs = 0.

Fig. 3
Fig. 3

Polar plot of widths vs scanning angle α for three different elliptical beam spots (solid curves). The spot for wa/wb = 2 is drawn as a dashed ellipse.

Fig. 4
Fig. 4

Test of a spot at z = 90 cm off the laser without intervening optics at 5 angles, 0°, 45°, 90°, 135°, and 180°. The fit w(α) = const results in w = (653 ± 4) μm. The sensitivity of the method to detect ellipticity is demonstrated by the dashed curve where tentatively Wb/wa = 0.9 has been set.

Fig. 5
Fig. 5

Goniometer to adjust the obliquity angle of the lens. The reference position is given by retroreflection which originally aligns the principal axis of the lens with the beam axis = optical axis.

Fig. 6
Fig. 6

Evolution of beam along the z axis. The circular Gaussian beam enters the lens at z = 0. The spots are shown as they appear in the x-y plane enlarged by ×100 relative to the z scale. Immediately to the left and right of the circle of least confusion, the tangential and sagittal waists are shown, respectively.

Fig. 7
Fig. 7

Scan of astigmatic beam at z = 60 cm from the lens at α = 45°. The least-squares fitted parameters are: total power P0 = (1.498 ± 0.005) mW, center at x ̅ s = ( 6.309 ± 0.002 ) mm, spot size w (45°) = (783 ± 6) μm.

Fig. 8
Fig. 8

Polar plot of spot sizes w(α), taken at z = 35 cm. The fit gives wa = (801 ± 3) μm, ε = 0.920 ± 0.003, α0 = (46.4 ± 0.4)°. The rms deviation of the widths in this plot from the best-fitted solid curve is 4.5 μm. The resulting elliptical spot size is drawn as a dashed curve.

Fig. 9
Fig. 9

Evolution of astigmatic beam vs z (see also Fig. 6). The waist sizes are w0t = (58.0 ± 0.2) μm and w0s = (102.0 ± 1.2) μm. They can be located with an accuracy of about ±2 mm.

Equations (19)

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E ( x , y ) = E 0 exp [ ( x 2 + y 2 ) / w 2 ]
S ( x , y ) = 2 π P 0 w a w b exp ( 2 x 2 / w a 2 ) exp ( 2 y 2 / w b 2 ) ,
x x 0 = x cos α 0 y sin α 0 , y y 0 = x sin α 0 + y cos α 0 .
P ( x s ) = x = x s x = + [ y = y = + S ( x , y ) d y ] d x .
P ( x s ) = ( P 0 / 2 ) erfc ( u ) , u = ( 2 ) ( x s x ̅ s ) / w ( α 0 ) ,
P ( x s ) / P 0 = Q ( υ ) , υ = [ 2 / w ( α 0 ) ] ( x s x ̅ s ) = ( 2 ) u ,
Q ( υ ) = [ 1 / ( 2 π ) ] υ exp ( t 2 / 2 ) d t ,
w 2 ( α 0 ) = w a 2 cos 2 α 0 + w b 2 sin 2 α 0 , x ̅ s ( α 0 ) = x 0 cos α 0 + y 0 sin α 0 + x s o
w 2 ( α ) = w a 2 cos 2 ( α 0 α ) + w b 2 sin 2 ( α 0 α ) , x ̅ s ( α ) = x 0 cos ( α 0 α ) + y 0 sin ( α 0 α ) + x s 0 .
υ + υ = 2 = [ 2 / w ( α ) ] ( x s + x ̅ s ) [ 2 / w ( α ) ] ( x s x ̅ s ) ,
w ( α ) = x s + x s .
P ( x ) = a 1 Q [ 2 ( x a 3 ) / a 2 ]
Q ( υ ) ½ + sgn ( υ ) { [ 1 / ( 2 π ) ] × ( a + b t + c t 2 ) t exp ( υ 2 / 2 ) 1 / 2 } , t = 1 / ( 1 + p | υ | ) , sgn ( υ < 0 ) = 1 , sgn ( υ = 0 ) = 0 , sgn ( υ > 0 ) = + 1 , p = 0.33267 , a = 0.4361836 , b = 0.1201676 , c = 0.9372980 .
w ( α ) = w a [ 1 ɛ 2 sin 2 ( α 0 α ) ] ,
w ( z ) = w 0 [ 1 + ( z z w ) 2 / z 0 2 ] ,
ϕ = arccos ( cos γ h cos γ υ ) .
α 0 = arcsin ( sin γ υ / sin ϕ ) .
[ P ( x s ) / P 0 ] TEM n 10 = ( 1 / 2 ) erfc ( u ) + ( u / π ) exp ( u 2 ) , u = ( 2 ) ( x x ̅ s ) / w .
Q(x){1+exp[f(x)]}1,f(x)=a1x+a2x3+a3x5+a4x7,a1=1.595700,a2=0.072953,a3=0.000324,a4=0.0000350.

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