Abstract

A modification in the analysis of a conventional laser beam spot size measurement method has been developed. The new analysis significantly decreases the uncertainty in the estimation of the beam-spot size. A conventional beam scanning approach was used in the measurement, but instead of differentiating the data and fitting the result to a Gaussian function, the data were fit to an analytical approximation to the complementary error function. As a result, fitted parameters were obtained that were consistent with the standard differentiation approach, but with considerably smaller uncertainty.

© 2006 Optical Society of America

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References

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  1. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964).
  2. N. G. Kingsbury, "Lecture notes on 'digital modulation,"' Courses I5 and 3F4, Department of Engineering, University of Cambridge, 1995-2005.

1995 (1)

N. G. Kingsbury, "Lecture notes on 'digital modulation,"' Courses I5 and 3F4, Department of Engineering, University of Cambridge, 1995-2005.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Kingsbury, N. G.

N. G. Kingsbury, "Lecture notes on 'digital modulation,"' Courses I5 and 3F4, Department of Engineering, University of Cambridge, 1995-2005.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Other (2)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964).

N. G. Kingsbury, "Lecture notes on 'digital modulation,"' Courses I5 and 3F4, Department of Engineering, University of Cambridge, 1995-2005.

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Figures (1)

Fig. 1
Fig. 1

(Color online) Points in (a) and (b) are laser beam spot size data taken using the translating knife-edge method. The dashed curves are fits to the complementary error function. The points in (c) and (d) are the derivatives of the data in (a) and (b), respectively. The dashed curves are fits to the Gaussians. (a) and (c) are for data containing 51 points, and (b) and (d) are for data containing 26 points over the same range.

Tables (1)

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Table 1 Summary of Results

Equations (13)

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I ( x , y ) = I 0 exp [ - 2 ( x - x 0 ) 2 w x 2 - 2 ( y - y 0 ) 2 w y 2 ] .
s ( x ) = x - I ( x , y ) d y d x = I 0 w y π / 2 x exp [ - 2 ( x - x 0 ) 2 w x 2 ] d x = I 0 π w x w y 2 erfc [ 2 ( x - x 0 ) w x ] = P erfc [ 2 ( x - x 0 ) w x ] .
erfc ( x ) = 1 2 π x exp ( - u 2 2 ) d u .
erf ( x ) = 1 - 2 erfc ( 2 x ) .
P ( R ) = 1 σ 2 π exp ( - R 2 2 σ 2 ) ,
S ( x ) = s ( x ) + R ( x ) .
d S ( x ) d x = d s ( x ) d x + d R ( x ) d x g ( x ) + D ( x ) ,
Pr [ R R i + 1 R + d R , R R i R + d R ] =  P ( R ) d R P ( R ) d R =  1 2 π σ 2 exp ( - R 2 + R 2 2 σ 2 ) d R 2 ,
P r [ D D i D + d D , R R i R + d R ] =  ϵ 2 π σ 2 exp [ - R 2 + ( R + ϵ D ) 2 2 σ 2 ] d R d D ,
P r [ D D i D + d D ] =  P ( D ) d D =  - ϵ 2 π σ 2 exp [ - R 2 + ( R + ϵ D ) 2 2 σ 2 ] d R d D =  1 σ 2 π exp ( - D 2 2 σ 2 ) d D ,
s ( x ) = P 2 { 1 - erf [ 2 ( x - x 0 ) w x ] } ,
erf ( x ) = 1 - ( a 1 t + a 2 t 2 + a 3 t 3 ) exp ( - x 2 ) + ϵ ( x ) , t = ( 1 + p x ) - 1 , p = 0.47047 , a 1 = 0.3480242 , a 2 = - 0.0958798 , a 3 = 0.7478556 ,
| ϵ ( x ) 2.5 × 10 - 5 | .

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