Abstract

We outline a theory for the calculation of the beam quality factor of an aberrated laser beam. We provide closed-form equations that show that the beam quality factor of an aberrated Gaussian beam depends on all primary aberrations except tilt, defocus, and x-astigmatism. The model is verified experimentally by implementing aberrations as digital holograms in the laboratory.

© 2011 Optical Society of America

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References

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  1. A. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
    [CrossRef]
  2. T. F. Johnston, Jr., “M2 concept characterizes beam quality,” Laser Focus World 26, 173–184 (May 1990).
  3. V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE, 1998).
  4. V. N. Mahajan, Optical Imaging and Aberrations, Part 2: Wave Diffraction Optics (SPIE, 1998).
  5. G-M. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).
    [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1998), pp. 517–553.
  7. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  8. A. Wada, H. Ohminato, T. Yonemura, Y. Miyamoto, and M. Takeda, “Effect of comatic aberration on the propagation characteristics of the Laguerre–Gaussian beams,” Opt. Rev. 12, 451–455 (2005).
    [CrossRef]
  9. O. Mendoza-Yero and J. Alda, “Irradiance map of an apertured Gaussian beam affected by coma,” Opt. Commun. 271, 517–523 (2007).
    [CrossRef]
  10. R. K. Singh, P. Senthilkumaran, and K. Singh, “The effect of astigmatism on the diffraction of a vortex carrying beam with a Gaussian background,” J. Opt. A 9, 543–554 (2007).
    [CrossRef]
  11. R. Borghi and M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
    [CrossRef] [PubMed]
  12. A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32, 5893–5091 (1993).
    [CrossRef] [PubMed]
  13. J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
    [CrossRef]
  14. J. Alda, J. Alonso, and E. Bernabeu, “Characterization of aberrated laser beams,” J. Opt. Soc. Am. A 14, 2737–2747 (1997).
    [CrossRef]
  15. T. A. Jeong and J. Lee, “Accurate determination of the beam quality factor of an aberrated high-power laser pulse,” J. Korean Phys. Soc. 55, 488–494 (2009).
    [CrossRef]
  16. R. Martinez-Herrero and P. M. Meijas, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18, 1669–1671 (1993).
    [CrossRef] [PubMed]
  17. C. Mafusire and A. Forbes, CSIR National Laser Centre, P.O. Box 395, Pretoria 0001, South Africa are preparing a manuscript to be called “Phase calibration of the Shack–Hartmann wavefront sensor using a phase only spatial sight modulator.”

2009 (1)

T. A. Jeong and J. Lee, “Accurate determination of the beam quality factor of an aberrated high-power laser pulse,” J. Korean Phys. Soc. 55, 488–494 (2009).
[CrossRef]

2007 (2)

O. Mendoza-Yero and J. Alda, “Irradiance map of an apertured Gaussian beam affected by coma,” Opt. Commun. 271, 517–523 (2007).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “The effect of astigmatism on the diffraction of a vortex carrying beam with a Gaussian background,” J. Opt. A 9, 543–554 (2007).
[CrossRef]

2005 (1)

A. Wada, H. Ohminato, T. Yonemura, Y. Miyamoto, and M. Takeda, “Effect of comatic aberration on the propagation characteristics of the Laguerre–Gaussian beams,” Opt. Rev. 12, 451–455 (2005).
[CrossRef]

1997 (2)

1994 (1)

J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
[CrossRef]

1993 (2)

1990 (1)

A. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

1976 (1)

Alda, J.

O. Mendoza-Yero and J. Alda, “Irradiance map of an apertured Gaussian beam affected by coma,” Opt. Commun. 271, 517–523 (2007).
[CrossRef]

J. Alda, J. Alonso, and E. Bernabeu, “Characterization of aberrated laser beams,” J. Opt. Soc. Am. A 14, 2737–2747 (1997).
[CrossRef]

Alonso, J.

Bernabeu, E.

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1998), pp. 517–553.

Dai, G-M.

G-M. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).
[CrossRef]

Forbes, A.

C. Mafusire and A. Forbes, CSIR National Laser Centre, P.O. Box 395, Pretoria 0001, South Africa are preparing a manuscript to be called “Phase calibration of the Shack–Hartmann wavefront sensor using a phase only spatial sight modulator.”

Jeong, T. A.

T. A. Jeong and J. Lee, “Accurate determination of the beam quality factor of an aberrated high-power laser pulse,” J. Korean Phys. Soc. 55, 488–494 (2009).
[CrossRef]

Johnston, T. F.

T. F. Johnston, Jr., “M2 concept characterizes beam quality,” Laser Focus World 26, 173–184 (May 1990).

Lee, J.

T. A. Jeong and J. Lee, “Accurate determination of the beam quality factor of an aberrated high-power laser pulse,” J. Korean Phys. Soc. 55, 488–494 (2009).
[CrossRef]

Mafusire, C.

C. Mafusire and A. Forbes, CSIR National Laser Centre, P.O. Box 395, Pretoria 0001, South Africa are preparing a manuscript to be called “Phase calibration of the Shack–Hartmann wavefront sensor using a phase only spatial sight modulator.”

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE, 1998).

V. N. Mahajan, Optical Imaging and Aberrations, Part 2: Wave Diffraction Optics (SPIE, 1998).

Martinez-Herrero, R.

Meijas, P. M.

Mendoza-Yero, O.

O. Mendoza-Yero and J. Alda, “Irradiance map of an apertured Gaussian beam affected by coma,” Opt. Commun. 271, 517–523 (2007).
[CrossRef]

Miyamoto, Y.

A. Wada, H. Ohminato, T. Yonemura, Y. Miyamoto, and M. Takeda, “Effect of comatic aberration on the propagation characteristics of the Laguerre–Gaussian beams,” Opt. Rev. 12, 451–455 (2005).
[CrossRef]

Noll, R. J.

Ohminato, H.

A. Wada, H. Ohminato, T. Yonemura, Y. Miyamoto, and M. Takeda, “Effect of comatic aberration on the propagation characteristics of the Laguerre–Gaussian beams,” Opt. Rev. 12, 451–455 (2005).
[CrossRef]

Ruff, J. A.

J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
[CrossRef]

Santarsiero, M.

Senthilkumaran, P.

R. K. Singh, P. Senthilkumaran, and K. Singh, “The effect of astigmatism on the diffraction of a vortex carrying beam with a Gaussian background,” J. Opt. A 9, 543–554 (2007).
[CrossRef]

Siegman, A.

A. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

Siegman, A. E.

J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
[CrossRef]

A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32, 5893–5091 (1993).
[CrossRef] [PubMed]

Singh, K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “The effect of astigmatism on the diffraction of a vortex carrying beam with a Gaussian background,” J. Opt. A 9, 543–554 (2007).
[CrossRef]

Singh, R. K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “The effect of astigmatism on the diffraction of a vortex carrying beam with a Gaussian background,” J. Opt. A 9, 543–554 (2007).
[CrossRef]

Takeda, M.

A. Wada, H. Ohminato, T. Yonemura, Y. Miyamoto, and M. Takeda, “Effect of comatic aberration on the propagation characteristics of the Laguerre–Gaussian beams,” Opt. Rev. 12, 451–455 (2005).
[CrossRef]

Wada, A.

A. Wada, H. Ohminato, T. Yonemura, Y. Miyamoto, and M. Takeda, “Effect of comatic aberration on the propagation characteristics of the Laguerre–Gaussian beams,” Opt. Rev. 12, 451–455 (2005).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1998), pp. 517–553.

Yonemura, T.

A. Wada, H. Ohminato, T. Yonemura, Y. Miyamoto, and M. Takeda, “Effect of comatic aberration on the propagation characteristics of the Laguerre–Gaussian beams,” Opt. Rev. 12, 451–455 (2005).
[CrossRef]

Appl. Opt. (1)

J. Korean Phys. Soc. (1)

T. A. Jeong and J. Lee, “Accurate determination of the beam quality factor of an aberrated high-power laser pulse,” J. Korean Phys. Soc. 55, 488–494 (2009).
[CrossRef]

J. Opt. A (1)

R. K. Singh, P. Senthilkumaran, and K. Singh, “The effect of astigmatism on the diffraction of a vortex carrying beam with a Gaussian background,” J. Opt. A 9, 543–554 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Laser Focus World (1)

T. F. Johnston, Jr., “M2 concept characterizes beam quality,” Laser Focus World 26, 173–184 (May 1990).

Opt. Commun. (1)

O. Mendoza-Yero and J. Alda, “Irradiance map of an apertured Gaussian beam affected by coma,” Opt. Commun. 271, 517–523 (2007).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
[CrossRef]

Opt. Rev. (1)

A. Wada, H. Ohminato, T. Yonemura, Y. Miyamoto, and M. Takeda, “Effect of comatic aberration on the propagation characteristics of the Laguerre–Gaussian beams,” Opt. Rev. 12, 451–455 (2005).
[CrossRef]

Proc. SPIE (1)

A. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

Other (5)

V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE, 1998).

V. N. Mahajan, Optical Imaging and Aberrations, Part 2: Wave Diffraction Optics (SPIE, 1998).

G-M. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).
[CrossRef]

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1998), pp. 517–553.

C. Mafusire and A. Forbes, CSIR National Laser Centre, P.O. Box 395, Pretoria 0001, South Africa are preparing a manuscript to be called “Phase calibration of the Shack–Hartmann wavefront sensor using a phase only spatial sight modulator.”

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Figures (4)

Fig. 1
Fig. 1

Added far-field divergence due to the aperture size as a result of truncation.

Fig. 2
Fig. 2

Beam quality factor of a truncated Gaussian beam for (a), (b)  γ = 0.3 , (c), (d)  γ = 1 , (e), (f)  γ = 2 , and (g), (h)  γ = 3 .

Fig. 3
Fig. 3

Experimental setup for the demonstration of optical aberrations in the laboratory. The digital holograms imparted to the laser beam are relay imaged to the wavefront sensor for an accurate phase measurement.

Fig. 4
Fig. 4

Beam quality factor dependence on rms Zernike primary aberrations (except tilt) in the x axis and y axis for the model (graphs) and the experimental results (data points).

Tables (3)

Tables Icon

Table 1 Names, Symbols, and Polynomials of the Zernike Primary Aberration Coefficients

Tables Icon

Table 2 Special Cases of the Beam Quality Factor Equations

Tables Icon

Table 3 Modular- 2 π Phase Screens of the Zernike Primary Aberrations

Equations (24)

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U ( ρ , θ ) = ψ ( ρ ) exp ( i ϕ ( ρ , θ ) ) ,
M x 2 = π λ ω x 2 θ x 2 V x 2 ,
M y 2 = π λ ω y 2 θ y 2 V y 2 ,
ω x 2 = 4 a 4 P 0 2 π 0 1 ψ 2 ρ 2 cos 2 θ ρ d ρ d θ ,
ω y 2 = 4 a 4 P 0 2 π 0 1 ψ 2 ρ 2 sin 2 θ ρ d ρ d θ ,
θ x 2 = λ 2 π 2 P 0 2 π 0 1 ( cos 2 θ ( d ψ d ρ ) 2 + ψ 2 ( cos θ ϕ ρ sin θ ρ ϕ θ ) 2 ) ρ d ρ d θ λ 2 a 2 π 2 P 2 ( 0 2 π 0 1 ψ 2 ( cos θ ϕ ρ sin θ ρ ϕ θ ) ρ d ρ d θ ) 2 + θ e 2 ,
θ y 2 = λ 2 π 2 P 0 2 π 0 1 ( sin 2 θ ( d ψ d ρ ) 2 + ψ 2 ( sin θ ϕ ρ + cos θ ρ ϕ θ ) 2 ) ρ d ρ d θ λ 2 a 2 π 2 P 2 ( 0 2 π 0 1 ψ 2 ( sin θ ϕ ρ + cos θ ρ ϕ θ ) ρ d ρ d θ ) 2 + θ e 2 .
V x = 2 λ a 2 π P 0 2 π 0 1 ψ 2 ρ cos θ ( cos θ ϕ ρ sin θ ρ ϕ θ ) ρ d ρ d θ ,
V y = 2 λ a 2 π P 0 2 π 0 1 ψ 2 ρ sin θ ( sin θ ϕ ρ + cos θ ρ ϕ θ ) ρ d ρ d θ .
θ e 2 = 16 λ 2 π 2 P ψ 2 ( 1 ) .
ϕ ( ρ , θ ) = 2 π n = 0 m A n 0 R n 0 ( ρ ) + 2 π n = 1 n = 1 m R n m ( ρ ) [ A n m cos m θ + B n m sin m θ ] .
U ( ρ , θ ) = ( 2 γ 2 π a 2 ) 1 / 4 exp ( γ 2 ρ 2 ) exp ( i ϕ ( ρ , θ ) ) , ρ 1 ,
θ e 2 = 32 γ 2 ( e 2 γ 2 1 ) π 3 ( λ a ) 2 .
M x 4 = 1 π [ exp ( 2 γ 2 ) 1 ] γ 8 { 24 π 3 [ B 22 2 [ exp ( 2 γ 2 ) 1 ] [ exp ( 2 γ 2 ) 2 γ 2 1 ] 2 γ 4 + 3 ( B 31 2 [ exp ( 2 γ 2 ) 1 ] [ exp ( 2 γ 2 ) 2 γ 2 1 ] [ exp ( 2 γ 2 ) 1 2 ( γ 4 + γ 2 ) ] γ 2 + A 31 2 [ exp ( 2 γ 2 ) 1 2 γ 2 ] [ 5 exp ( 4 γ 2 ) + 2 γ 2 ( γ 2 + 1 ) 2 exp ( 2 γ 2 ) ( 9 γ 4 + γ 2 + 5 ) + 5 ] γ 2 + 2 ( B 31 B 33 + A 31 A 33 ) [ exp ( 2 γ 2 ) 1 ] [ exp ( 2 γ 2 ) 1 2 γ 2 ] [ exp ( 2 γ 2 ) 2 ( γ 4 + γ 2 ) 1 ] γ 2 + [ exp ( 2 γ 2 ) 1 ] { ( B 33 2 + A 33 2 ) [ exp ( 2 γ 2 ) 1 2 γ 2 ] [ exp ( 2 γ 2 ) 1 2 ( γ 4 + γ 2 ) ] γ 2 + 20 A 40 2 [ 2 γ 2 ( γ 2 + 2 ) + exp ( 4 γ 2 ) 2 exp ( 2 γ 2 ) ( 2 γ 6 γ 4 + 2 γ 2 + 1 ) + 1 ] } ) ] + [ exp ( 2 γ 2 ) 1 ] [ exp ( 2 γ 2 ) 1 2 γ 2 ] { π [ exp ( 2 γ 2 ) 1 2 γ 2 ] + 32 } γ 8 } ,
M y 4 = 1 π [ exp ( 2 γ 2 ) 1 ] γ 8 { 24 π 3 [ B 22 2 [ exp ( 2 γ 2 ) 1 ] [ exp ( 2 γ 2 ) 1 2 γ 2 ] 2 γ 4 + 3 ( B 31 2 ( exp ( 2 γ 2 ) 1 2 γ 2 ] [ 5 exp ( 4 γ 2 ) + 2 γ 2 ( γ 2 + 1 ) 2 exp ( 2 γ 2 ) ( 9 γ 4 + γ 2 + 5 ) + 5 ] γ 2 + 2 B 31 B 33 [ exp ( 2 γ 2 ) 1 ] { exp ( 2 γ 2 ) 1 2 γ 2 } [ exp ( 2 γ 2 ) 1 2 ( γ 4 + γ 2 ) ] γ 2 + [ exp ( 2 γ 2 ) 1 ] ( exp ( 4 γ 2 ) { [ B 33 2 + ( A 31 A 33 ) 2 ] γ 2 + 20 A 40 2 } + γ 2 { 40 A 40 2 ( γ 2 + 2 ) + [ B 33 2 + ( A 31 A 33 ) 2 ] [ 4 γ 6 + 6 γ 4 + 4 γ 2 + 1 ] } 2 exp ( 2 γ 2 ) { [ B 33 2 + ( A 31 A 33 ) 2 ] [ γ 3 + γ ] 2 + 20 A 40 2 ( 2 γ 6 γ 4 + 2 γ 2 + 1 ) ] + 20 A 40 2 } ) ] + [ exp ( 2 γ 2 ) 1 ] [ exp ( 2 γ 2 ) 1 2 γ 2 ] [ π ( exp ( 2 γ 2 ) 1 2 γ 2 ) + 32 ] γ 8 } ,
0 2 π cos m θ cos q θ d θ = π ( 1 + δ m 0 ) δ m q , 0 2 π sin m θ sin q θ d θ = π ( 1 δ m 0 ) δ m q , 0 2 π cos m θ sin q θ d θ = 0 , and 0 1 R n m ( ρ ) R p q ( ρ ) ρ d ρ = δ n p δ m q 2 ( n + 1 ) .
U ( x , y ) = ψ ( x , y ) exp ( i ϕ ( x , y ) ) ,
M x 2 = 4 π λ x 2 0 D x 2 ,
x s D x t = a s + 2 P ψ 2 ( x x ) s ( D x D x ) t d x d y ,
x 2 = a 4 P ψ 2 x 2 d x d y .
D x = λ 2 π ψ a | U x | = λ 2 π ψ a ( ψ x ) 2 + ( ψ ϕ x ) 2 .
D x 2 = a 2 1 P ψ 2 ( D x D x ) 2 d x d y = λ 2 4 π 2 ( a 2 P ( ( ψ x ) 2 + ( ψ ϕ x ) 2 ) d x d y ( a P ψ 2 ϕ x d x d y ) 2 ) .
M x 2 = 4 π λ x 2 D x 2 x D x 2 ,
x D x = a 3 P ψ 2 x D x d x d y = λ 2 π a 2 P ψ 2 x ϕ x d x d y .

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