Abstract

A method for approximating the inverse error function involved in the determination of the radius of a Gaussian beam is proposed. It is based on a polynomial inversion that can be developed to any desired degree, according to an a priori defined error budget. Analytic expressions are obtained and used to determine the radius of a TEMoo He–Ne laser beam from intensity measurements experimentally obtained by using the knife edge method. The error and the interval of validity of the approximation are determined for polynomials of different degrees. The analysis of the theoretical and experimental errors is also presented.

© 2013 Optical Society of America

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References

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  1. P. J. Shayler, “Laser beam distribution in the focal region,” Appl. Opt. 17, 2673–2674 (1978).
    [CrossRef]
  2. Y. Suzaki and A. Tachibana, “Measurement of the μm sized radius of Gaussian laser beam using the scanning knife-edge,” Appl. Opt. 14, 2809–2810 (1975).
    [CrossRef]
  3. M. S. Scholl, “Measured spatial properties of the CW Nd:YAG laser beam,” Appl. Opt. 19, 3655–3659 (1980).
    [CrossRef]
  4. A. K. Cherri and M. S. Alam, “Accurate measurement of small Gaussian laser beam diameters using various rulings,” Opt. Commun. 223, 255–262 (2003).
    [CrossRef]
  5. R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt. 23, 2227 (1984).
    [CrossRef]
  6. A. Rose, Y.-X. Nie, and R. Gupta, “Laser beam profile measurement by photothermal deflection technique,” Appl. Opt. 25, 1738–1741 (1986).
    [CrossRef]
  7. T. Baba, T. Arai, and A. Ono, “Laser beam profile measurement by a thermographic technique,” Rev. Sci. Instrum. 57, 2739–2742 (1986).
    [CrossRef]
  8. A. E. Siegman, M. W. Sasnett, and T. F. Johnston, “Choice of clip levels for beam width measurements using knife-edge techniques,” IEEE J. Quantum Electron. 27, 1098–1104 (1991).
    [CrossRef]
  9. R. Díaz-Uribe, M. Rosete-Aguilar, and R. Ortega-Martínez, “Position sensing of a Gaussian beam with a power meter and knife edge,” Rev. Mex. Fís. 39, 484–492 (1992).
  10. G. Veshapidze, M. L. Trachy, M. H. Shah, and B. D. De Paola, “Reducing the uncertainty in laser beam size measurement with a scanning edge method,” Appl. Opt. 45, 8197–8199 (2006).
    [CrossRef]
  11. J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. 22, 3406–3410 (1983).
    [CrossRef]
  12. M. A. C. de Araujo, R. Silva, E. de Lima, D. P. Pereira, and P. C. de Oliveira, “Measurement of Gaussian laser beam radius using the knife-edge technique: improvement on data analysis,” Appl. Opt. 48, 393–396 (2009).
    [CrossRef]
  13. G. Arfken, Mathematical Methods for Physicists (Academic, 1981).
  14. B. H. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (Macmillan, 1961), p. 136.
  15. M. F. González-Cardel, “Determinación de los coeficientes de asfericidad de una superficie óptica rápida (aspheric coefficients determination for a fast optical surface),” M. Sc. dissertation (Facultad de Ciencias, UNAM, México, 2003).
  16. M. F. González-Cardel and R. Díaz-Uribe, “An analysis on the inversion of polynomials,” Rev. Mex. Fís. E 52, 163–171 (2006).
  17. Wolfram Research, “Inverse error function,” ( http://functions.wolfram.com/GammaBetaErf/InverseErf/06/01/01/01/0001/ ).
  18. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef]
  19. D. C. Baird, Experimentation: An Introduction to Measurement Theory and Experimental Design (Prentice Hall, 1962).
  20. M. Spivak, Calculus (W. A. Benjamín, 1967), pp. 345–352.

2009 (1)

2006 (2)

G. Veshapidze, M. L. Trachy, M. H. Shah, and B. D. De Paola, “Reducing the uncertainty in laser beam size measurement with a scanning edge method,” Appl. Opt. 45, 8197–8199 (2006).
[CrossRef]

M. F. González-Cardel and R. Díaz-Uribe, “An analysis on the inversion of polynomials,” Rev. Mex. Fís. E 52, 163–171 (2006).

2003 (1)

A. K. Cherri and M. S. Alam, “Accurate measurement of small Gaussian laser beam diameters using various rulings,” Opt. Commun. 223, 255–262 (2003).
[CrossRef]

1992 (1)

R. Díaz-Uribe, M. Rosete-Aguilar, and R. Ortega-Martínez, “Position sensing of a Gaussian beam with a power meter and knife edge,” Rev. Mex. Fís. 39, 484–492 (1992).

1991 (1)

A. E. Siegman, M. W. Sasnett, and T. F. Johnston, “Choice of clip levels for beam width measurements using knife-edge techniques,” IEEE J. Quantum Electron. 27, 1098–1104 (1991).
[CrossRef]

1986 (2)

T. Baba, T. Arai, and A. Ono, “Laser beam profile measurement by a thermographic technique,” Rev. Sci. Instrum. 57, 2739–2742 (1986).
[CrossRef]

A. Rose, Y.-X. Nie, and R. Gupta, “Laser beam profile measurement by photothermal deflection technique,” Appl. Opt. 25, 1738–1741 (1986).
[CrossRef]

1984 (1)

1983 (1)

1980 (1)

1978 (1)

1975 (1)

1966 (1)

Alam, M. S.

A. K. Cherri and M. S. Alam, “Accurate measurement of small Gaussian laser beam diameters using various rulings,” Opt. Commun. 223, 255–262 (2003).
[CrossRef]

Arai, T.

T. Baba, T. Arai, and A. Ono, “Laser beam profile measurement by a thermographic technique,” Rev. Sci. Instrum. 57, 2739–2742 (1986).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, 1981).

Baba, T.

T. Baba, T. Arai, and A. Ono, “Laser beam profile measurement by a thermographic technique,” Rev. Sci. Instrum. 57, 2739–2742 (1986).
[CrossRef]

Baird, D. C.

D. C. Baird, Experimentation: An Introduction to Measurement Theory and Experimental Design (Prentice Hall, 1962).

Cherri, A. K.

A. K. Cherri and M. S. Alam, “Accurate measurement of small Gaussian laser beam diameters using various rulings,” Opt. Commun. 223, 255–262 (2003).
[CrossRef]

de Araujo, M. A. C.

de Lima, E.

de Oliveira, P. C.

De Paola, B. D.

Díaz-Uribe, R.

M. F. González-Cardel and R. Díaz-Uribe, “An analysis on the inversion of polynomials,” Rev. Mex. Fís. E 52, 163–171 (2006).

R. Díaz-Uribe, M. Rosete-Aguilar, and R. Ortega-Martínez, “Position sensing of a Gaussian beam with a power meter and knife edge,” Rev. Mex. Fís. 39, 484–492 (1992).

Dwight, B. H.

B. H. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (Macmillan, 1961), p. 136.

Garetz, B. A.

González-Cardel, M. F.

M. F. González-Cardel and R. Díaz-Uribe, “An analysis on the inversion of polynomials,” Rev. Mex. Fís. E 52, 163–171 (2006).

M. F. González-Cardel, “Determinación de los coeficientes de asfericidad de una superficie óptica rápida (aspheric coefficients determination for a fast optical surface),” M. Sc. dissertation (Facultad de Ciencias, UNAM, México, 2003).

Gupta, R.

Johnston, T. F.

A. E. Siegman, M. W. Sasnett, and T. F. Johnston, “Choice of clip levels for beam width measurements using knife-edge techniques,” IEEE J. Quantum Electron. 27, 1098–1104 (1991).
[CrossRef]

Khosrofian, J. M.

Kogelnik, H.

Li, T.

McCally, R. L.

Nie, Y.-X.

Ono, A.

T. Baba, T. Arai, and A. Ono, “Laser beam profile measurement by a thermographic technique,” Rev. Sci. Instrum. 57, 2739–2742 (1986).
[CrossRef]

Ortega-Martínez, R.

R. Díaz-Uribe, M. Rosete-Aguilar, and R. Ortega-Martínez, “Position sensing of a Gaussian beam with a power meter and knife edge,” Rev. Mex. Fís. 39, 484–492 (1992).

Pereira, D. P.

Rose, A.

Rosete-Aguilar, M.

R. Díaz-Uribe, M. Rosete-Aguilar, and R. Ortega-Martínez, “Position sensing of a Gaussian beam with a power meter and knife edge,” Rev. Mex. Fís. 39, 484–492 (1992).

Sasnett, M. W.

A. E. Siegman, M. W. Sasnett, and T. F. Johnston, “Choice of clip levels for beam width measurements using knife-edge techniques,” IEEE J. Quantum Electron. 27, 1098–1104 (1991).
[CrossRef]

Scholl, M. S.

Shah, M. H.

Shayler, P. J.

Siegman, A. E.

A. E. Siegman, M. W. Sasnett, and T. F. Johnston, “Choice of clip levels for beam width measurements using knife-edge techniques,” IEEE J. Quantum Electron. 27, 1098–1104 (1991).
[CrossRef]

Silva, R.

Spivak, M.

M. Spivak, Calculus (W. A. Benjamín, 1967), pp. 345–352.

Suzaki, Y.

Tachibana, A.

Trachy, M. L.

Veshapidze, G.

Appl. Opt. (9)

IEEE J. Quantum Electron. (1)

A. E. Siegman, M. W. Sasnett, and T. F. Johnston, “Choice of clip levels for beam width measurements using knife-edge techniques,” IEEE J. Quantum Electron. 27, 1098–1104 (1991).
[CrossRef]

Opt. Commun. (1)

A. K. Cherri and M. S. Alam, “Accurate measurement of small Gaussian laser beam diameters using various rulings,” Opt. Commun. 223, 255–262 (2003).
[CrossRef]

Rev. Mex. Fís. (1)

R. Díaz-Uribe, M. Rosete-Aguilar, and R. Ortega-Martínez, “Position sensing of a Gaussian beam with a power meter and knife edge,” Rev. Mex. Fís. 39, 484–492 (1992).

Rev. Mex. Fís. E (1)

M. F. González-Cardel and R. Díaz-Uribe, “An analysis on the inversion of polynomials,” Rev. Mex. Fís. E 52, 163–171 (2006).

Rev. Sci. Instrum. (1)

T. Baba, T. Arai, and A. Ono, “Laser beam profile measurement by a thermographic technique,” Rev. Sci. Instrum. 57, 2739–2742 (1986).
[CrossRef]

Other (6)

Wolfram Research, “Inverse error function,” ( http://functions.wolfram.com/GammaBetaErf/InverseErf/06/01/01/01/0001/ ).

D. C. Baird, Experimentation: An Introduction to Measurement Theory and Experimental Design (Prentice Hall, 1962).

M. Spivak, Calculus (W. A. Benjamín, 1967), pp. 345–352.

G. Arfken, Mathematical Methods for Physicists (Academic, 1981).

B. H. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (Macmillan, 1961), p. 136.

M. F. González-Cardel, “Determinación de los coeficientes de asfericidad de una superficie óptica rápida (aspheric coefficients determination for a fast optical surface),” M. Sc. dissertation (Facultad de Ciencias, UNAM, México, 2003).

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Figures (7)

Fig. 1.
Fig. 1.

Plot to calculate the solution interval of the function ε(χ) [see Eq. (11)].

Fig. 2.
Fig. 2.

Plot of the limits of the interval of validity for different degrees of the approximating polynomial.

Fig. 3.
Fig. 3.

Experimental device.

Fig. 4.
Fig. 4.

Plot of optical power versus position, where the S-shaped curve is observed.

Fig. 5.
Fig. 5.

Simulated and experimental laser beam intensity distribution curve versus the knife-edge position at 54.7 cm. Dashed lines represent the valid interval in which the series expansion is valid.

Fig. 6.
Fig. 6.

Calculated beam width of simulated data using Eqs. (4) and (6). We can observe that there exists an indeterminate value, which appears when the intensity is a half of its maximum intensity.

Fig. 7.
Fig. 7.

Plot that shows the radius of beams for all experimental data, with approximations to different degrees.

Tables (2)

Tables Icon

Table 1. Intervals of Validity for Every Approximation

Tables Icon

Table 2. Radii Average

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

I(x,y)=Ioexp[2(xxo)2+(yyo)2ro2],
P(x)=PT2[1+erf(2(xxo)ro)],
erf(z)=2π0zet2dt.
ro=2(xxo)erf1(2PPT1),
r08π(xxo)(2PPT1).
ro2πPT(x2x1)(P2P1)=2πPTm,
erf(χ)=2πn=0(1)nχ2n+1n!(2n+1)=n=0a2n+1χ2n+1.
erf(χ)2π{χχ31!(3)}ξ.
χ(ξ)=erf1(ξ)π2ξ+π324ξ3.
r08πxξ{1+π12ξ2}1,
ε(χ)|T1χ3+T2χ4++T6χ8|e1000,
T1=5a235a1a2a3a13,T2=6a24+a1a22a33a12a32a14,T3=2a25+11a1a23a37a12a2a32a15,T4=6a24a3+3a1a22a323a12a33a15,T5=6a23a323a1a2a33a15,T6=2a22a33a1a34a15.
a2n+1=2π(1)nn!(2n+1),
T2=3(a3a1)2=13T4=3(a3a1)3=19,T6=(a3a1)4=181.
χ2N(ξ)=erf1(ξ)π2ξn=0Ns2nξ2n,
ro8πxξ{n=0Ns2nξ2n}1.
w=w01+(λzπw02)2,
R2n+1(ξ)=2π|ξ2n+3n!(2n+3)|.
δf|2πξ2n+3n!(2n+3)2π{i=0n(1)iξ2i+1i!(2i+1)}|=|ξ2n+3n!(2n+3){i=0n(1)iξ2i+1i!(2i+1)}|.
δx=5×107mx,
δP=1μWP,
δrP=7π5/296ξ5+π3/24ξ3+πξ7π5/2960ξ5+π3/224ξ3+π2ξδP,
δrx=(2{7π5/2960{2PPT1}5+π3/224{2PPT1}3+π2{2PPT1}})δx,
δr=0.01+δrx+δrP+δf,
δr=0.01+|1χ2n(ξ)|{|i=0(n1)/2(2z2i+1)(2i)!(d(2i+1)dz(2i+1)χ2n(ξ)|z=0)|δP+2δx+ξ2n+3(2n+3)n!},
χ2n(ξ)=2π{i=0n(1)iξ2i+1i!(2i+1)},ξ=2PPT1.

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