Abstract

The limits of the paraxial approximation for a laser beam under ABCD transformations is established through the relationship between a parameter concerning the beam paraxiality, the paraxial estimator, and the beam second-order moments. The applicability of such an estimator is extended to an optical system composed by optical elements as mirrors and lenses and sections of free space, what completes the analysis early performed for free-space propagation solely. As an example, the paraxiality of a system composed by free space and a spherical thin lens under the propagation of Hermite-Gauss and Laguerre-Gauss modes is established. The results show that the the paraxial approximation fails for a certain feasible range of values of main parameters. In this sense, the paraxial estimator is an useful tool to monitor the limits of the paraxial optics theory under ABCD transformations.

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  1. R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical system and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
    [CrossRef] [PubMed]
  2. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
    [CrossRef] [PubMed]
  3. A. E. Siegman, Lasers, (University Science Books, 1986), Chaps. 15–21.
  4. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (John Wiley, 1991), Chaps. 1–4,7–9,14.
    [CrossRef]
  5. H. Kogelnik, “Imaging of Optical Modes -Resonators and Internal Lenses”, Bell Syst. Opt. Tech. J. 44, 455–494 (1965).
  6. P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial aproximation in laser beams,” Opt. Lett. 32, 927–929 (2007).
    [CrossRef] [PubMed]
  7. P. Vaveliuk, “Comment on degree of paraxiality for monochromatic light beams,” Opt. Lett. 33, 3004–3005 (2008).
    [CrossRef] [PubMed]
  8. P. Vaveliuk, G. F. Zebende, M. A. Moret, and B. Ruiz, “Propagating free-space nonparaxial beams,” J. Opt. Soc. Am. A 24, 3297–3302 (2007).
    [CrossRef]
  9. M. A. Bandres and M. Guizar-Sicairos, “Paraxial group,” Opt. Lett. 34, 13–15 (2009).
    [CrossRef]
  10. S. R. Seshadri, “Quality of paraxial electromagnetic beams,” Appl. Opt. 45, 5335–5345 (2006).
    [CrossRef] [PubMed]
  11. P. Vaveliuk, “Quantifying the paraxiality for laser beams from the M2-factor,” Opt. Lett. 34, 340–342 (2009).
    [CrossRef] [PubMed]
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  13. K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
    [CrossRef]
  14. G. Nemes and A. E. Siegmam, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11, 2257–2264 (1994).
    [CrossRef]
  15. A. E. Siegman, G. Nemes, and J. Serna, in Proceedings of DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.
  16. S. Ramee and R. Simon, “Effect of holes and vortices on beam quality,” J. Opt. Soc. Am. A 17, 84–94 (2000).
    [CrossRef]
  17. M. Nazarathy and J. Shamir, “First-order optics–a canonical operator representation lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
    [CrossRef]
  18. E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
    [CrossRef]
  19. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
    [CrossRef]
  20. H. T. Yura, B. Rose, and S. G. Hanson, “Dynamic laser speckle in complex ABCD optical systems” J. Opt. Soc. Am. A 15, 1160–1166 (1998).
    [CrossRef]

2009 (2)

2008 (1)

2007 (2)

2006 (1)

2000 (1)

1998 (1)

1995 (1)

1994 (1)

1991 (1)

1987 (1)

1985 (2)

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical system and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

1982 (1)

1965 (1)

H. Kogelnik, “Imaging of Optical Modes -Resonators and Internal Lenses”, Bell Syst. Opt. Tech. J. 44, 455–494 (1965).

Bandres, M. A.

Bélanger, P. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Guizar-Sicairos, M.

Hanson, S. G.

Kogelnik, H.

H. Kogelnik, “Imaging of Optical Modes -Resonators and Internal Lenses”, Bell Syst. Opt. Tech. J. 44, 455–494 (1965).

Lencina, A.

Moret, M. A.

Mukunda, N.

K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical system and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Nazarathy, M.

Nemes, G.

G. Nemes and A. E. Siegmam, “Measurement of all ten second-order moments of an astigmatic beam by the use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am. A 11, 2257–2264 (1994).
[CrossRef]

A. E. Siegman, G. Nemes, and J. Serna, in Proceedings of DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

Ramee, S.

Rose, B.

Ruiz, B.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (John Wiley, 1991), Chaps. 1–4,7–9,14.
[CrossRef]

Serna, J.

A. E. Siegman, G. Nemes, and J. Serna, in Proceedings of DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

Seshadri, S. R.

Shamir, J.

Siegmam, A. E.

Siegman, A. E.

A. E. Siegman, Lasers, (University Science Books, 1986), Chaps. 15–21.

A. E. Siegman, G. Nemes, and J. Serna, in Proceedings of DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

Simon, R.

S. Ramee and R. Simon, “Effect of holes and vortices on beam quality,” J. Opt. Soc. Am. A 17, 84–94 (2000).
[CrossRef]

K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12, 560–569 (1995).
[CrossRef]

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical system and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical system and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Sundar, K.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (John Wiley, 1991), Chaps. 1–4,7–9,14.
[CrossRef]

Vaveliuk, P.

Yura, H. T.

Zebende, G. F.

Appl. Opt. (1)

Bell Syst. Opt. Tech. J. (1)

H. Kogelnik, “Imaging of Optical Modes -Resonators and Internal Lenses”, Bell Syst. Opt. Tech. J. 44, 455–494 (1965).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Acta (1)

E. C. G. Sudarshan, N. Mukunda, and R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. A (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical system and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Other (4)

A. E. Siegman, G. Nemes, and J. Serna, in Proceedings of DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), paper MQ1.

A. E. Siegman, Lasers, (University Science Books, 1986), Chaps. 15–21.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, (John Wiley, 1991), Chaps. 1–4,7–9,14.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Figures (3)

Fig. 1
Fig. 1

Illustration of several ABCD system crossed by a light beam of waist size wm. The systems begin at the beam waist plane (z = 0) and end at the output plane (z = z′). (a) free space propagation over a distance d. (b) spherical thin lens (of negligible thickness δ) and focal length f. (c) free space plus spherical thin lens.

Fig. 2
Fig. 2

A density plot of the paraxial estimator (paraxial-nonparaxial map) (a) for free space and (b) spherical thin lens systems crossed by a fundamental mode (N = 0) in terms of the normalized beam waist parameter 0. The vertical axis for (b) represents the focal length of the lens f (normalized to λ). There is no parameter in the vertical axis for (a). This axis was maintained with the purpose to compare both system results.

Fig. 3
Fig. 3

A density plot of ��′ as a function of 0 and d/f for the system depicted in Fig. 1(c) crossed by the mode N = 0. The dotted line is a reference for the paraxial-nonparaxial limit in the input plane, with value �� = 0.94 that was included for comparative purposes.

Equations (23)

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( u , v ) = E ( x , y ) e 2 π i ( u x + v y ) d x d y ,
W ( r , p ) = E ( r + 1 2 r ) E * ( r 1 2 r ) exp ( i 2 π r p ) d r ,
m s s = s s W ( r , p ) d r d p W ( r , p ) d r d p ,
M = [ 𝕄 q q 𝕄 q p 𝕄 q p t 𝕄 p p ] = [ m x x m x y m x u m x v m y x m y y m y u m y v m u x m u y m u u m u v m v x m v y m v u m v v ] ,
T = [ 𝔸 𝔹 𝔻 ] ,
T Ω T t = Ω , Ω = [ 𝕆 𝕀 𝕀 𝕆 ] ,
M = T M T t
𝒫 = 1 λ 2 2 ( m u u + m v v ) .
𝒫 = 1 λ 2 2 ( m u u + m v v ) = 1 λ 2 2 tr 𝕄 p p ,
𝕄 p p = 𝕄 q q t + 𝕄 q p 𝔻 t + 𝔻 𝕄 p q t + 𝔻 𝕄 p p 𝔻 t .
E m , n ( x , y , 0 ) = E 0 H m ( 2 x w 0 ) H n ( 2 y w 0 ) exp ( x 2 + y 2 w 0 2 )
E , p ( r , ϕ , 0 ) = E 0 2 / 2 L p ( 2 r 2 w 0 2 ) exp ( i ϕ ) exp ( r 2 w 0 2 )
M H G = 1 4 [ ( 2 m + 1 ) w 0 2 0 0 0 0 ( 2 n + 1 ) w 0 2 0 0 0 0 ( 2 m + 1 ) π 2 w 0 2 0 0 0 0 ( 2 n + 1 ) π 2 w 0 2 ] .
T F = [ 𝕀 λ d 𝕀 𝕆 𝕀 ] ,
M = T F M T F = [ 𝕄 q q + d λ ( 𝕄 q p + 𝕄 q p t ) + ( d λ ) 2 𝕄 p p 𝕄 q p + z λ 𝕄 p p 𝕄 q p t + d λ 𝕄 p p 𝕄 p p ] .
𝒫 = 𝒫 = 1 ( N + 1 ) 4 π 2 w ˜ 0 2 ,
T L = [ 𝕀 𝕆 𝔾 𝕀 ] ,
𝔾 = [ g x g x y g x y g y ] ,
𝕄 p p = 𝕄 p p 𝔾 𝕄 q p ( 𝔾 𝕄 q p ) t + 𝔾 𝕄 q q 𝔾 .
𝒫 = 𝒫 ( N + 1 ) w 0 2 4 f 2 ,
𝒫 1 1 4 F # 2 ,
𝕄 p p = 𝕄 p p 𝔾 ( 𝕄 q p + d λ 𝕄 p p ) ( 𝕄 q p t + z λ 𝕄 p p ) 𝔾 + 𝔾 [ 𝕄 q q + d λ ( 𝕄 q p + 𝕄 q p t ) + ( d λ ) 2 𝕄 p p ] 𝔾 .
𝒫 = 1 ( N + 1 ) [ ( d / f 1 ) 2 4 π 2 w ˜ 0 2 + w 0 2 4 f 2 ] .

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