Abstract

We outline an approach for the calculation of the mean focal length of an aberrated lens and provide closed-form solutions that show that the focal length of the lens is dependent on the presence of defocus, x-astigmatism, and spherical aberration. The results are applicable to Gaussian beams in the presence of arbitrary-sized apertures. The theoretical results are confirmed experimentally, showing excellent agreement. As the final results are in algebraic form, the theory may readily be applied in the laboratory if the aberration coefficients of the lens are known.

© 2011 Optical Society of America

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References

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  1. V. N. Mahajan, Aberration Theory Made Simple (SPIE, 1991).
    [CrossRef]
  2. V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE, 1998).
  3. F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1976).
  4. A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32, 5893-5091(1993).
    [CrossRef] [PubMed]
  5. 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]
  6. W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381-2395 (2001).
    [CrossRef]
  7. J. H. Sung, T. M. Jeong, S. K. Lee, T. J. Yu, I. W. Choi, and J. Lee, “Analysis of thermal aberrations in the power amplifiers of a 10 Hz 100 TW Ti:sapphire laser,” J. Korean Phys. Soc. 55, 495-500 (2009).
    [CrossRef]
  8. N. Hodgson and H. Weber, Resonator Fundamentals: Fundamentals, Advanced Concepts and Applications (Springer-Verlag, 1997), Ch. 13 and Ch. 15.
    [PubMed]
  9. W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, 1999), Fig. 6.82, p. 389.
  10. P. A. Belanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196-198 (1991).
    [CrossRef] [PubMed]
  11. A. E. Siegman, “Defining the effective radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146-1148 (1991).
    [CrossRef]
  12. J. Alda, J. Alonso, and E. Bernabeu, “Characterization of aberrated laser beams,” J. Opt. Soc. Am. A 14, 2737-2747 (1997).
    [CrossRef]
  13. 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.
  14. G-M. Dai, Wavefront Optics for Vision Correction (SPIE, 2008).
    [CrossRef]
  15. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]
  16. The term (1+δm0) prefixing Eq. has been added to correct for an error in .
  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 light modulator.”
  18. A. Montmerle-Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersee, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
    [CrossRef]
  19. M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modelling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831-1833 (1990).
    [CrossRef]
  20. J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode at high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
    [CrossRef]
  21. S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state lasers: the case of ytterbium-doped materials,” Prog. Quantum Electron. 30, 89-153 (2006).
    [CrossRef]
  22. A. Forbes and L. R. Botha, “Physical optics modelling of intra-cavity thermal distortions in solid-state laser resonators,” Proc. SPIE 4768, 153-163 (2002).
    [CrossRef]
  23. E. H. Bernhardi, A. Forbes, C. Bollig, and M. J. D. Esser, “Estimation of thermal fracture limits in quasi-continuous-wave end-pumped through a time-dependent analytical model,” Opt. Express 16, 11115-11123 (2008).
    [CrossRef] [PubMed]
  24. D. C. Hanna, C. G. Sawyers, and M. A. Yuratich, “Telescopic resonators for large-volume TEM00-mode operation,” Opt. Quantum Electron. 13, 493-507 (1981).
    [CrossRef]

2009 (1)

J. H. Sung, T. M. Jeong, S. K. Lee, T. J. Yu, I. W. Choi, and J. Lee, “Analysis of thermal aberrations in the power amplifiers of a 10 Hz 100 TW Ti:sapphire laser,” J. Korean Phys. Soc. 55, 495-500 (2009).
[CrossRef]

2008 (1)

2006 (2)

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state lasers: the case of ytterbium-doped materials,” Prog. Quantum Electron. 30, 89-153 (2006).
[CrossRef]

A. Montmerle-Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersee, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
[CrossRef]

2002 (2)

A. Forbes and L. R. Botha, “Physical optics modelling of intra-cavity thermal distortions in solid-state laser resonators,” Proc. SPIE 4768, 153-163 (2002).
[CrossRef]

J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode at high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
[CrossRef]

2001 (1)

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381-2395 (2001).
[CrossRef]

1997 (1)

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 (1)

1991 (2)

P. A. Belanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196-198 (1991).
[CrossRef] [PubMed]

A. E. Siegman, “Defining the effective radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146-1148 (1991).
[CrossRef]

1990 (1)

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modelling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831-1833 (1990).
[CrossRef]

1981 (1)

D. C. Hanna, C. G. Sawyers, and M. A. Yuratich, “Telescopic resonators for large-volume TEM00-mode operation,” Opt. Quantum Electron. 13, 493-507 (1981).
[CrossRef]

1976 (1)

Alda, J.

Alonso, J.

Balembois, F.

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state lasers: the case of ytterbium-doped materials,” Prog. Quantum Electron. 30, 89-153 (2006).
[CrossRef]

Belanger, P. A.

Bernabeu, E.

Bernhardi, E. H.

Bollig, C.

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.

Botha, L. R.

A. Forbes and L. R. Botha, “Physical optics modelling of intra-cavity thermal distortions in solid-state laser resonators,” Proc. SPIE 4768, 153-163 (2002).
[CrossRef]

Bourderionnet, J.

J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode at high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
[CrossRef]

Brignon, A.

J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode at high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
[CrossRef]

Chénais, S.

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state lasers: the case of ytterbium-doped materials,” Prog. Quantum Electron. 30, 89-153 (2006).
[CrossRef]

Choi, I. W.

J. H. Sung, T. M. Jeong, S. K. Lee, T. J. Yu, I. W. Choi, and J. Lee, “Analysis of thermal aberrations in the power amplifiers of a 10 Hz 100 TW Ti:sapphire laser,” J. Korean Phys. Soc. 55, 495-500 (2009).
[CrossRef]

Clarkson, W. A.

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381-2395 (2001).
[CrossRef]

Dai, G-M.

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

Druon, F.

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state lasers: the case of ytterbium-doped materials,” Prog. Quantum Electron. 30, 89-153 (2006).
[CrossRef]

Esser, M. J. D.

Fields, R. A.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modelling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831-1833 (1990).
[CrossRef]

Fincher, C. L.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modelling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831-1833 (1990).
[CrossRef]

Forbes, A.

E. H. Bernhardi, A. Forbes, C. Bollig, and M. J. D. Esser, “Estimation of thermal fracture limits in quasi-continuous-wave end-pumped through a time-dependent analytical model,” Opt. Express 16, 11115-11123 (2008).
[CrossRef] [PubMed]

A. Forbes and L. R. Botha, “Physical optics modelling of intra-cavity thermal distortions in solid-state laser resonators,” Proc. SPIE 4768, 153-163 (2002).
[CrossRef]

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 light modulator.”

Forget, S.

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state lasers: the case of ytterbium-doped materials,” Prog. Quantum Electron. 30, 89-153 (2006).
[CrossRef]

Frey, R.

J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode at high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
[CrossRef]

Georges, P.

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state lasers: the case of ytterbium-doped materials,” Prog. Quantum Electron. 30, 89-153 (2006).
[CrossRef]

Gilbert, M.

A. Montmerle-Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersee, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
[CrossRef]

Hanna, D. C.

D. C. Hanna, C. G. Sawyers, and M. A. Yuratich, “Telescopic resonators for large-volume TEM00-mode operation,” Opt. Quantum Electron. 13, 493-507 (1981).
[CrossRef]

Hodgson, N.

N. Hodgson and H. Weber, Resonator Fundamentals: Fundamentals, Advanced Concepts and Applications (Springer-Verlag, 1997), Ch. 13 and Ch. 15.
[PubMed]

Huignard, J.-P.

J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode at high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
[CrossRef]

Innocenzi, M. E.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modelling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831-1833 (1990).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1976).

Jeong, T. M.

J. H. Sung, T. M. Jeong, S. K. Lee, T. J. Yu, I. W. Choi, and J. Lee, “Analysis of thermal aberrations in the power amplifiers of a 10 Hz 100 TW Ti:sapphire laser,” J. Korean Phys. Soc. 55, 495-500 (2009).
[CrossRef]

Koechner, W.

W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, 1999), Fig. 6.82, p. 389.

Lee, J.

J. H. Sung, T. M. Jeong, S. K. Lee, T. J. Yu, I. W. Choi, and J. Lee, “Analysis of thermal aberrations in the power amplifiers of a 10 Hz 100 TW Ti:sapphire laser,” J. Korean Phys. Soc. 55, 495-500 (2009).
[CrossRef]

Lee, S. K.

J. H. Sung, T. M. Jeong, S. K. Lee, T. J. Yu, I. W. Choi, and J. Lee, “Analysis of thermal aberrations in the power amplifiers of a 10 Hz 100 TW Ti:sapphire laser,” J. Korean Phys. Soc. 55, 495-500 (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 light modulator.”

Mahajan, V. N.

V. N. Mahajan, Aberration Theory Made Simple (SPIE, 1991).
[CrossRef]

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

Montmerle-Bonnefois, A.

A. Montmerle-Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersee, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
[CrossRef]

Noll, R.

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]

Sawyers, C. G.

D. C. Hanna, C. G. Sawyers, and M. A. Yuratich, “Telescopic resonators for large-volume TEM00-mode operation,” Opt. Quantum Electron. 13, 493-507 (1981).
[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]

A. E. Siegman, “Defining the effective radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146-1148 (1991).
[CrossRef]

Sung, J. H.

J. H. Sung, T. M. Jeong, S. K. Lee, T. J. Yu, I. W. Choi, and J. Lee, “Analysis of thermal aberrations in the power amplifiers of a 10 Hz 100 TW Ti:sapphire laser,” J. Korean Phys. Soc. 55, 495-500 (2009).
[CrossRef]

Thro, P.-Y.

A. Montmerle-Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersee, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
[CrossRef]

Weber, H.

N. Hodgson and H. Weber, Resonator Fundamentals: Fundamentals, Advanced Concepts and Applications (Springer-Verlag, 1997), Ch. 13 and Ch. 15.
[PubMed]

Weulersee, J.-M.

A. Montmerle-Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersee, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
[CrossRef]

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1976).

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.

Yu, T. J.

J. H. Sung, T. M. Jeong, S. K. Lee, T. J. Yu, I. W. Choi, and J. Lee, “Analysis of thermal aberrations in the power amplifiers of a 10 Hz 100 TW Ti:sapphire laser,” J. Korean Phys. Soc. 55, 495-500 (2009).
[CrossRef]

Yura, H. T.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modelling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831-1833 (1990).
[CrossRef]

Yuratich, M. A.

D. C. Hanna, C. G. Sawyers, and M. A. Yuratich, “Telescopic resonators for large-volume TEM00-mode operation,” Opt. Quantum Electron. 13, 493-507 (1981).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modelling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831-1833 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. E. Siegman, “Defining the effective radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146-1148 (1991).
[CrossRef]

J. Korean Phys. Soc. (1)

J. H. Sung, T. M. Jeong, S. K. Lee, T. J. Yu, I. W. Choi, and J. Lee, “Analysis of thermal aberrations in the power amplifiers of a 10 Hz 100 TW Ti:sapphire laser,” J. Korean Phys. Soc. 55, 495-500 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. D (1)

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381-2395 (2001).
[CrossRef]

Opt. Commun. (2)

J. Bourderionnet, A. Brignon, J.-P. Huignard, and R. Frey, “Influence of aberrations on fundamental mode at high power rod solid-state lasers,” Opt. Commun. 204, 299-310 (2002).
[CrossRef]

A. Montmerle-Bonnefois, M. Gilbert, P.-Y. Thro, and J.-M. Weulersee, “Thermal lensing and spherical aberration in high-power transversally pumped laser rods,” Opt. Commun. 259, 223-235 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (2)

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]

D. C. Hanna, C. G. Sawyers, and M. A. Yuratich, “Telescopic resonators for large-volume TEM00-mode operation,” Opt. Quantum Electron. 13, 493-507 (1981).
[CrossRef]

Proc. SPIE (1)

A. Forbes and L. R. Botha, “Physical optics modelling of intra-cavity thermal distortions in solid-state laser resonators,” Proc. SPIE 4768, 153-163 (2002).
[CrossRef]

Prog. Quantum Electron. (1)

S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state lasers: the case of ytterbium-doped materials,” Prog. Quantum Electron. 30, 89-153 (2006).
[CrossRef]

Other (9)

The term (1+δm0) prefixing Eq. has been added to correct for an error in .

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 light modulator.”

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.

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

V. N. Mahajan, Aberration Theory Made Simple (SPIE, 1991).
[CrossRef]

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

F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, 1976).

N. Hodgson and H. Weber, Resonator Fundamentals: Fundamentals, Advanced Concepts and Applications (Springer-Verlag, 1997), Ch. 13 and Ch. 15.
[PubMed]

W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, 1999), Fig. 6.82, p. 389.

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Figures (5)

Fig. 1
Fig. 1

Spherically aberrated lens focusing the rays in the margins with the paraxial rays focused further out.

Fig. 2
Fig. 2

Experimental setup to create an aberrated lens, in the form of a digital hologram, in the laboratory, and the subsequent measurement of the focal length thereof.

Fig. 3
Fig. 3

Curvature dependence on (a) defocus, (b) spherical aberration, (c) x-astigmatism and (d) the rest of the primary aberrations. The theoretical curve for the x and y axes is determined from Eqs. (21a, 21b). The background aberrations included in the theoretical predictions were (a)  A 22 = 0.0123384 λ and A 40 = 0.0009048 λ , (b)  A 20 = 0.00050279 λ and A 22 = 0.00123384 λ , and (c)  A 20 = 0.0050279 λ and A 40 = 0.0209048 λ .

Fig. 4
Fig. 4

Fractional change in focal power, Δ D , of a defocus aberrated lens compared with one with spherical aberration. The defocus coefficient was 0.1 wave of A 20 .

Fig. 5
Fig. 5

Calibration curve for the phase-only SLM. LUT, look-up table.

Tables (4)

Tables Icon

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

Tables Icon

Table 2 Some Special Cases of the Mean Focal Length of an Aberrated Lens

Tables Icon

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

Tables Icon

Table 4 Gradient of the Curves in Fig. 3

Equations (29)

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

1 R out = 1 R in + 1 f ,
f = R out .
U ( x , y ) = ψ ( x , y ) exp [ i ϕ ( x , y ) ] ,
1 R x = λ 2 π 1 1 1 1 x ϕ x ψ 2 ( x , y ) d x d y a 2 1 1 1 1 x 2 ψ 2 ( x , y ) d x d y , 1 R y = λ 2 π 1 1 1 1 y ϕ y ψ 2 ( x , y ) d x d y a 2 1 1 1 1 y 2 ψ 2 ( x , y ) d x d y .
f x = 2 π λ ( 1 1 1 1 x ϕ x ψ 2 ( x , y ) d x d y a 2 1 1 1 1 x 2 ψ 2 ( x , y ) d x d y ) 1 , f y = 2 π λ ( 1 1 1 1 y ϕ y ψ 2 ( x , y ) d x d y a 2 1 1 1 1 y 2 ψ 2 ( x , y ) d x d y ) 1 .
ϕ ( x ) = k x 2 2 f 0 ,
x ϕ x ψ 2 = k f 0 x 2 ψ 2 ,
ϕ ( ρ , θ ) = 2 π n = 0 A n 0 R n 0 ( ρ ) + 2 π n = 1 m = 1 n R n m ( ρ ) [ A n m cos m θ + B n m sin m θ ] ,
R n m ( ρ ) = k = 0 n m 2 ( 1 ) k ( n k ) ! ρ n 2 k k ! ( n + m 2 k ) ! ( n m 2 k ) ! ,
A n m = 1 π 2 ( n + 1 ) 1 + δ m 0 0 2 π 0 1 ρ d ρ d θ ϕ ( ρ , θ ) R n m ( ρ ) cos m θ ,
B n m = 1 π 2 ( n + 1 ) 1 + δ m 0 0 2 π 0 1 ρ d ρ d θ ϕ ( ρ , θ ) R n m ( ρ ) sin m θ .
ϕ ( ρ , θ ) = k a 2 ρ 2 2 f ,
A 20 = A 00 = 3 a 2 12 λ f .
f = 3 a 2 12 λ A 20 .
f x = 2 π 2 λ ( 0 2 π 0 1 ρ 2 cos θ ( cos θ ϕ ( ρ , θ ) ρ sin θ ρ ϕ ( ρ , θ ) θ ) ψ 2 ( ρ ) d ρ d θ a 2 0 1 ρ 3 ψ 2 ( ρ ) d ρ ) 1 ,
f y = 2 π 2 λ ( 0 2 π 0 1 ρ 2 sin θ ( sin θ ϕ ( ρ , θ ) ρ + cos θ ρ ϕ ( ρ , θ ) θ ) ψ 2 ( ρ ) d ρ d θ a 2 0 1 ρ 3 ψ 2 ( ρ ) d ρ ) 1 .
f = 2 π λ ( 0 1 ρ 2 d ϕ ( ρ ) d ρ ψ 2 ( ρ ) d ρ a 2 0 1 ρ 3 ψ 2 ( ρ ) d ρ ) 1 .
U ( ρ , θ ) = ( 2 γ 2 π a 2 ) 1 / 4 exp ( γ 2 ρ 2 ) exp ( i ϕ ( ρ , θ ) ) .
f ± = π a 2 0 1 ρ 3 ψ 2 ( ρ ) d ρ 2 λ ( 2 π n = 0 A n 0 0 1 ρ 2 d R n 0 ( ρ ) d ρ ψ 2 ( ρ ) d ρ + n = 1 m = 1 n 0 2 π 0 1 ρ 2 ( 1 ± cos 2 θ ) d R n m ( ρ ) d ρ [ A n m cos m θ + B n m sin m θ ] ψ 2 ( ρ ) d ρ d θ n = 1 m = 1 n 0 2 π 0 1 ρ 2 sin 2 θ R n m ( ρ ) d d θ [ A n m cos m θ + B n m sin m θ ] ψ 2 ( ρ ) d ρ d θ ) 1 ,
d R n m ( ρ ) d ρ = ( 1 + δ m 0 ) n [ R n 1 , m 1 ( ρ ) + R n 1 , m + 1 ( ρ ) ] + d R n 2 , m ( ρ ) d ρ ,
0 1 ρ k 1 exp ( 2 γ 2 ρ 2 ) d ρ = 1 2 ( 2 γ 2 ) 1 2 k [ Γ ( 1 2 k ) Γ ( 1 2 k , 2 γ 2 ) ] ,
f = a 2 2 λ ( 2 3 A 20 6 A 22 + 6 5 ( 1 2 γ 2 + 4 γ 2 e 2 γ 2 1 2 γ 2 ) A 40 ) 1 .
f x = a 2 2 λ ( 2 3 A 20 6 A 22 + 6 5 ( 1 2 γ 2 + 4 γ 2 e 2 γ 2 1 2 γ 2 ) A 40 ) 1 ,
f y = a 2 2 λ ( 2 3 A 20 + 6 A 22 + 6 5 ( 1 2 γ 2 + 4 γ 2 e 2 γ 2 1 2 γ 2 ) A 40 ) 1 .
ϕ = A r 2 + B r 4 + C .
f 1 = 1 2 λ A .
f 2 = 1 2 λ ( 2 a 2 B γ 2 ( 2 γ 4 e 2 γ 2 2 γ 2 1 ) A ) 1 .
spherical aberration , 1 f = 1 R 12 5 A 40 a 2 ;
astigmatism , 1 f = 1 R 2 6 A 22 a 2 .

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