Abstract

The fundamental Gaussian TEM00 mode is the most common mode of propagation within various optical devices, modules, and systems. Beam profilers are widely used in accurately ascertaining the cross-sectional irradiance profile of a TEM00 mode for free-space optical communication systems as well as tracking beam evolution when propagating within optical submodules. We demonstrate beam profiling methods that use low-cost, off-the-shelf, widely available circular apertures such as circular irises and spatial filters. In order to demonstrate beam profiling with any circular aperture, we first derive exact analytical expressions for power transmittance of the TEM00 mode through a decentered circular aperture and then use this mathematical derivation to estimate the irradiance profile of a Gaussian beam by 1) fixing the location of a circular aperture and changing its radius, and 2) scanning the entire area of the beam profile by translating a circular aperture of a fixed radius across the region of interest. This method is fast and easily reproducible and simply puts to use circular irises/circular spatial filters, which are commonly available in most optical laboratories. Consequently, the proposed method provides cheap and convenient means to estimate the profile of a Gaussian beam with simple optical components. Our experimental results demonstrate a performance that is comparable to a standard knife-edge-based estimate of beam profile. Moreover, a strong agreement with presented theory validates the analytical expressions derived in this paper.

© 2019 Optical Society of America

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References

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  1. S. A. Reza, T. S. Khwaja, M. A. Mazhar, H. K. Niazi, and R. Nawab, “Improved laser-based triangulation sensor with enhanced range and resolution through adaptive optics-based active beam control,” Appl. Opt. 56, 5996–6006 (2017).
    [Crossref]
  2. S. A. Reza and A. Anjum, “Robust motion-free and error-correcting method of estimating the focal length of a lens,” Appl. Opt. 56, 342–353 (2017).
    [Crossref]
  3. S. Yuan and N. A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses,” Appl. Opt. 38, 3214–3222 (1999).
    [Crossref]
  4. M. Qasim and S. A. Reza, “Toward the design of a motion-free tunable coupling module for varying spatial beam profiles: foundations of optimal coupling of a Gaussian mode into a fiber collimator with a dynamic two-lens system,” Appl. Opt. 54, 9242–9252 (2015).
    [Crossref]
  5. D. R. Skinner and R. E. Whitcher, “Measurement of the radius of a high-power laser beam near the focus of a lens,” J. Phys. E 5, 237–238 (1972).
    [Crossref]
  6. 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]
  7. M. A. de Araújo, 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]
  8. S. Sumriddetchkajorn and N. A. Riza, “Micro-electro-mechanical system-based digitally controlled optical beam profiler,” Appl. Opt. 41, 3506–3510 (2002).
    [Crossref]
  9. M. Gentili and N. A. Riza, “Wide-aperture no-moving-parts optical beam profiler using liquid-crystal displays,” Appl. Opt. 46, 506–512 (2007).
    [Crossref]
  10. P. J. Shayler, “Laser beam distribution in the focal region,” Appl. Opt. 17, 2673–2674 (1978).
    [Crossref]
  11. M. Sheikh and N. A. Riza, “Demonstration of pinhole laser beam profiling using a digital micromirror device,” IEEE Photon. Technol. Lett. 21, 666–668 (2009).
    [Crossref]
  12. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. De la Claviere, E. A. Franke, and J. M. Franke, “Technique for fast measurement of Gaussian laser beam parameters,” Appl. Opt. 10, 2775–2776 (1971).
    [Crossref]
  13. J. T. Knudtson and K. L. Ratzlaff, “Laser beam spatial profile analysis using a two‐dimensional photodiode array,” Rev. Sci. Instrum. 54, 856–860 (1983).
    [Crossref]
  14. Thorlabs, “M2MS-BC106N Operation Manual, Version 7.0,” 2017.
  15. F. W. Sheu and C. H. Chang, “Measurement of the intensity profile of a Gaussian laser beam near its focus using an optical fiber,” Am. J. Phys. 75, 956–959 (2007).
    [Crossref]
  16. J. M. Liu, “Simple technique for measurements of pulsed Gaussian-beam spot sizes,” Opt. Lett. 7, 196–198 (1982).
    [Crossref]
  17. N. A. Riza and M. A. Mazhar, “Laser beam imaging via multiple mode operations of the extreme dynamic range CAOS camera,” Appl. Opt. 57, E20–E31 (2018).
    [Crossref]
  18. Thorlabs, SM2D25D—SM2 Ring-Actuated Iris Diaphragm.
  19. N. R. Barbeau, “Power deposited by a Gaussian beam on a decentered circular aperture,” Appl. Opt. 34, 6443–6445 (1995).
    [Crossref]
  20. M. Abramowitz and I. A. Stegun, “Bessel functions of integer order,” in Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, 0009 revised ed. (Dover, 1972), pp. 374–377.
  21. G. B. Arfken and H. J. Weber, “The gamma function (factorial function),“ in Mathematical Methods for Physicists, 6th ed. (Academic, 2005), pp. 565–572.
  22. Thorlabs, “S120C Datasheet, Compact Photodiode Power Head with Silicon Detector,” 18356-S01, Rev C (2016).

2018 (1)

2017 (2)

2015 (1)

2009 (2)

M. A. de Araújo, 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]

M. Sheikh and N. A. Riza, “Demonstration of pinhole laser beam profiling using a digital micromirror device,” IEEE Photon. Technol. Lett. 21, 666–668 (2009).
[Crossref]

2007 (2)

F. W. Sheu and C. H. Chang, “Measurement of the intensity profile of a Gaussian laser beam near its focus using an optical fiber,” Am. J. Phys. 75, 956–959 (2007).
[Crossref]

M. Gentili and N. A. Riza, “Wide-aperture no-moving-parts optical beam profiler using liquid-crystal displays,” Appl. Opt. 46, 506–512 (2007).
[Crossref]

2002 (1)

1999 (1)

1995 (1)

1983 (1)

J. T. Knudtson and K. L. Ratzlaff, “Laser beam spatial profile analysis using a two‐dimensional photodiode array,” Rev. Sci. Instrum. 54, 856–860 (1983).
[Crossref]

1982 (1)

1978 (1)

1975 (1)

1972 (1)

D. R. Skinner and R. E. Whitcher, “Measurement of the radius of a high-power laser beam near the focus of a lens,” J. Phys. E 5, 237–238 (1972).
[Crossref]

1971 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, “Bessel functions of integer order,” in Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, 0009 revised ed. (Dover, 1972), pp. 374–377.

Anjum, A.

Arfken, G. B.

G. B. Arfken and H. J. Weber, “The gamma function (factorial function),“ in Mathematical Methods for Physicists, 6th ed. (Academic, 2005), pp. 565–572.

Arnaud, J. A.

Barbeau, N. R.

Chang, C. H.

F. W. Sheu and C. H. Chang, “Measurement of the intensity profile of a Gaussian laser beam near its focus using an optical fiber,” Am. J. Phys. 75, 956–959 (2007).
[Crossref]

de Araújo, M. A.

De la Claviere, B.

de Lima, E.

de Oliveira, P. C.

Franke, E. A.

Franke, J. M.

Gentili, M.

Hubbard, W. M.

Khwaja, T. S.

Knudtson, J. T.

J. T. Knudtson and K. L. Ratzlaff, “Laser beam spatial profile analysis using a two‐dimensional photodiode array,” Rev. Sci. Instrum. 54, 856–860 (1983).
[Crossref]

Liu, J. M.

Mandeville, G. D.

Mazhar, M. A.

Nawab, R.

Niazi, H. K.

Pereira, D. P.

Qasim, M.

Ratzlaff, K. L.

J. T. Knudtson and K. L. Ratzlaff, “Laser beam spatial profile analysis using a two‐dimensional photodiode array,” Rev. Sci. Instrum. 54, 856–860 (1983).
[Crossref]

Reza, S. A.

Riza, N. A.

Shayler, P. J.

Sheikh, M.

M. Sheikh and N. A. Riza, “Demonstration of pinhole laser beam profiling using a digital micromirror device,” IEEE Photon. Technol. Lett. 21, 666–668 (2009).
[Crossref]

Sheu, F. W.

F. W. Sheu and C. H. Chang, “Measurement of the intensity profile of a Gaussian laser beam near its focus using an optical fiber,” Am. J. Phys. 75, 956–959 (2007).
[Crossref]

Silva, R.

Skinner, D. R.

D. R. Skinner and R. E. Whitcher, “Measurement of the radius of a high-power laser beam near the focus of a lens,” J. Phys. E 5, 237–238 (1972).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, “Bessel functions of integer order,” in Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, 0009 revised ed. (Dover, 1972), pp. 374–377.

Sumriddetchkajorn, S.

Suzaki, Y.

Tachibana, A.

Weber, H. J.

G. B. Arfken and H. J. Weber, “The gamma function (factorial function),“ in Mathematical Methods for Physicists, 6th ed. (Academic, 2005), pp. 565–572.

Whitcher, R. E.

D. R. Skinner and R. E. Whitcher, “Measurement of the radius of a high-power laser beam near the focus of a lens,” J. Phys. E 5, 237–238 (1972).
[Crossref]

Yuan, S.

Am. J. Phys. (1)

F. W. Sheu and C. H. Chang, “Measurement of the intensity profile of a Gaussian laser beam near its focus using an optical fiber,” Am. J. Phys. 75, 956–959 (2007).
[Crossref]

Appl. Opt. (12)

N. R. Barbeau, “Power deposited by a Gaussian beam on a decentered circular aperture,” Appl. Opt. 34, 6443–6445 (1995).
[Crossref]

S. Yuan and N. A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses,” Appl. Opt. 38, 3214–3222 (1999).
[Crossref]

S. Sumriddetchkajorn and N. A. Riza, “Micro-electro-mechanical system-based digitally controlled optical beam profiler,” Appl. Opt. 41, 3506–3510 (2002).
[Crossref]

M. Gentili and N. A. Riza, “Wide-aperture no-moving-parts optical beam profiler using liquid-crystal displays,” Appl. Opt. 46, 506–512 (2007).
[Crossref]

J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. De la Claviere, E. A. Franke, and J. M. Franke, “Technique for fast measurement of Gaussian laser beam parameters,” Appl. Opt. 10, 2775–2776 (1971).
[Crossref]

P. J. Shayler, “Laser beam distribution in the focal region,” Appl. Opt. 17, 2673–2674 (1978).
[Crossref]

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]

M. A. de Araújo, 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]

M. Qasim and S. A. Reza, “Toward the design of a motion-free tunable coupling module for varying spatial beam profiles: foundations of optimal coupling of a Gaussian mode into a fiber collimator with a dynamic two-lens system,” Appl. Opt. 54, 9242–9252 (2015).
[Crossref]

S. A. Reza and A. Anjum, “Robust motion-free and error-correcting method of estimating the focal length of a lens,” Appl. Opt. 56, 342–353 (2017).
[Crossref]

S. A. Reza, T. S. Khwaja, M. A. Mazhar, H. K. Niazi, and R. Nawab, “Improved laser-based triangulation sensor with enhanced range and resolution through adaptive optics-based active beam control,” Appl. Opt. 56, 5996–6006 (2017).
[Crossref]

N. A. Riza and M. A. Mazhar, “Laser beam imaging via multiple mode operations of the extreme dynamic range CAOS camera,” Appl. Opt. 57, E20–E31 (2018).
[Crossref]

IEEE Photon. Technol. Lett. (1)

M. Sheikh and N. A. Riza, “Demonstration of pinhole laser beam profiling using a digital micromirror device,” IEEE Photon. Technol. Lett. 21, 666–668 (2009).
[Crossref]

J. Phys. E (1)

D. R. Skinner and R. E. Whitcher, “Measurement of the radius of a high-power laser beam near the focus of a lens,” J. Phys. E 5, 237–238 (1972).
[Crossref]

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

J. T. Knudtson and K. L. Ratzlaff, “Laser beam spatial profile analysis using a two‐dimensional photodiode array,” Rev. Sci. Instrum. 54, 856–860 (1983).
[Crossref]

Other (5)

Thorlabs, “M2MS-BC106N Operation Manual, Version 7.0,” 2017.

Thorlabs, SM2D25D—SM2 Ring-Actuated Iris Diaphragm.

M. Abramowitz and I. A. Stegun, “Bessel functions of integer order,” in Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, 0009 revised ed. (Dover, 1972), pp. 374–377.

G. B. Arfken and H. J. Weber, “The gamma function (factorial function),“ in Mathematical Methods for Physicists, 6th ed. (Academic, 2005), pp. 565–572.

Thorlabs, “S120C Datasheet, Compact Photodiode Power Head with Silicon Detector,” 18356-S01, Rev C (2016).

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Figures (7)

Fig. 1.
Fig. 1. Incident Gaussian beam (a) centered and (b) decentered at a circular aperture of radius “a.”
Fig. 2.
Fig. 2. Beam profiling with a fixed circular aperture location and varying aperture radius.
Fig. 3.
Fig. 3. Beam profiling by translating a circular aperture of a fixed radius.
Fig. 4.
Fig. 4. Experimental setup for beam profiling with circular apertures.
Fig. 5.
Fig. 5. Selected displayed images on DMD for expanding and scanning aperture schemes for beam location 2.
Fig. 6.
Fig. 6. Data points and estimated best-fit curve for the first beam location with (a) expanding circular aperture, (b) scanning circular aperture, and (c) knife-edge technique.
Fig. 7.
Fig. 7. Data points and estimated best-fit curve for the second beam location with (a) expanding circular aperture, (b) scanning circular aperture, and (c) knife-edge technique.

Tables (3)

Tables Icon

Table 1. Measurement Data of First Beam with a) Varying Circular Aperture Radius, b) Changing Circular Aperture Location, and c) Moving Knife Edge

Tables Icon

Table 2. Measurement Data of Second Beam with a) Varying Circular Aperture Radius, b) Changing Circular Aperture Location, and c) Moving Knife-Edge

Tables Icon

Table 3. Measured R2 Values of Each Experimental Dataset Verifying the Validity of the Proposed Theoretical Expressions

Equations (18)

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

I(x,y,z)=IPeak(w0w(z))2exp(2((x)2+(y)2)w2(z)).
I(x,y,z=z)=IPeak(w0w)2exp(2(x2+y2)w2)×exp(2(dx2+dy2)w2)exp(4xdxw2)exp(4ydyw2).
I(ρ,θ,z)=IPeak(w0w)2exp(2ρ2w2)exp(2d2w2)×exp(4ρdxcosθw2)exp(4ρdysinθw2).
PT=0a02πIPeak(w0w)2exp(2ρ2w2)exp(2d2w2)×exp(4ρdxcosθw2)exp(4ρdysinθw2)dθρdρ.
PT=IPeak(w0w)2exp(2d2w2)×2π0aρexp(2ρ2w2)I0(4ρdw2)dρ.
PT=IPeak(w0w)2exp(2d2w2)×2π0aρexp(2ρ2w2)k=04kd2kw4k(k!)2ρ2kdρ,
PT=IPeak(w0w)2exp(2d2w2)×2πk=04kd2kw4k(k!)20aρ2k+1exp(2ρ2w2)dρ.
PT=IPeak(w0w)2exp(2d2w2)2πk=04kd2kw4k(k!)2×[a2k2k2w2(a2w2)k(Γ(k+1)Γ(k+1,2a2w2))].
PT=πw02IPeak2exp(2d2w2)×k=02kd2kw2k(k!)2(Γ(k+1)Γ(k+1,2a2w2)),
PT=πw02IPeak2exp(2d2w2)k=02kd2kw2k(k!)2(γ(k+1,2a2w2)).
γ(k+1,2a2w2)=Γ(k+1)Γ(k+1,2a2w2)γ(k+1,2a2w2)=k!(1exp(2a2w2)i=0k1i!(2a2w2)i).
PT=πw02IPeak2exp(2d2w2)×k=0[2kd2kw2kk!(1exp(2a2w2)i=0k2ia2iw2ii!)].
PTotal=002πIPeak(w0w)2exp(2ρ2w2)dθρdρ=πw02IPeak2.
T=PTPTotal=exp(2d2w2)×k=0[2kd2kw2kk!(1exp(2a2w2)i=0k2ia2iw2ii!)].
d2k=0fork0,d2k=1fork=0.
Td=0=PT(d=0)Ptotal=(1exp(2a2w2)).
T=erf[2x0w],
R2=1i(yifi)2i(yiy¯)2,

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