Abstract

We studied the influence of thermal birefringence-induced bifocusing on optical beams. An analytical model was developed to describe the bifocusing aberration. With this model we determined the circle-of-least-confusion location and the amount of beam quality degradation expected. A numerical model was developed to describe wave-front deformation by a bifocal lens and the intensity distribution at the foci of the polarization components—radial and tangential as well as at the circle of least confusion. Model predictions for beam quality degradation and intensity distributions along the focal range were confirmed experimentally using a strongly pumped Nd:YAG rod as a bifocal lens. It is shown that bifocusing is an important aberration in high-power rod-based lasers that must (and can) be corrected to achieve high beam quality.

© 2005 Optical Society of America

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References

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  1. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, Reading, Mass., 1962).
  2. W. Koechner, "Thermal lensing in a Nd:YAG laser rod," Appl. Opt. 9, 2548-2553 (1970).
    [CrossRef] [PubMed]
  3. J. D. Foster and L. M. Osterink, "Thermal effects in a Nd:YAG laser," J. Appl. Phys. 41, 3656-3663 (1970).
    [CrossRef]
  4. W. Koechner and D. K. Rice, "Effect of birefringence on the performance of linearly polarized YAG:Nd lasers," IEEE J. Quantum Electron. QE-6, 557-566 (1970).
    [CrossRef]
  5. W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, New York, 1999).
    [CrossRef]
  6. J. T. Verdeyen, Laser Electronics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N. Y., 1981).
  7. P. A. Bélanger, "Beam propagation and the ABCD ray matrices," Opt. Lett. 16, 196-198 (1991).
    [CrossRef]
  8. A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D.A.Holmes, ed., Proc. SPIE 1224, 2-14 (1990).
    [CrossRef]
  9. S. Lavi, R. Prochaska, and E. Keren, "Generalized beam parameters and transformation laws for partially coherent light," Appl. Opt. 27, 3696-3703 (1988).
    [CrossRef] [PubMed]
  10. Q. Lü, S. Dong, and H. Weber, "Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod," Opt. Quantum Electron. 27, 777-783 (1995).
    [CrossRef]
  11. I. Moshe, S. Jackel, and A. Meir, "Production of radially or tangentially polarized beams in solid-state lasers and elimination of thermally induced birefringence effects," Opt. Lett. 28, 807-809 (2003).
    [CrossRef] [PubMed]
  12. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  13. I. Moshe, S. Jackel, and R. Lallouz, "Working beyond the static limits of laser stability by use of adaptive and polarization-conjugation optics," Appl. Opt. 37, 6415-6420 (1998).
    [CrossRef]
  14. W. C. Scott and M. de Wit, "Birefringence compensation and TEM00 mode enhancement in a Nd:YAG laser," Appl. Phys. Lett. 18, 3-4 (1971).
    [CrossRef]
  15. Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, "A novel approach for compensation of birefringence in cylindrical Nd:YAG rods," Opt. Quantum Electron. 28, 57-69 (1996).
    [CrossRef]
  16. G. Giuliani and P. Ristori, "Polarization flip cavities: a new approach to laser resonators," Opt. Commun. 35, 109-112 (1980).
    [CrossRef]
  17. S. Jackel, I. Moshe, A. Kaufman, A. Lavi, and R. Lallouz, "High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifier," Opt. Eng. 36, 2031-2036 (1997).
    [CrossRef]
  18. I. Moshe and S. Jackel, "Correction of birefringence and thermal lensing in nonreciprocal resonators by use of a dynamic imaging mirror," Appl. Opt. 39, 4313-4319 (2000).
    [CrossRef]
  19. I. Moshe and S. Jackel, "Correction of thermally induced birefringence in double-rod laser resonators--comparison of various methods," Opt. Commun. 214, 315-325 (2002).
    [CrossRef]
  20. I. Moshe, S. Jackel, and A. Meir, "Beam quality improvement in thermally birefringent Nd:YAG laser amplifiers by use of radially polarized beams," in Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2004).

2003 (1)

2002 (1)

I. Moshe and S. Jackel, "Correction of thermally induced birefringence in double-rod laser resonators--comparison of various methods," Opt. Commun. 214, 315-325 (2002).
[CrossRef]

2000 (1)

1998 (1)

1997 (1)

S. Jackel, I. Moshe, A. Kaufman, A. Lavi, and R. Lallouz, "High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifier," Opt. Eng. 36, 2031-2036 (1997).
[CrossRef]

1996 (1)

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, "A novel approach for compensation of birefringence in cylindrical Nd:YAG rods," Opt. Quantum Electron. 28, 57-69 (1996).
[CrossRef]

1995 (1)

Q. Lü, S. Dong, and H. Weber, "Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod," Opt. Quantum Electron. 27, 777-783 (1995).
[CrossRef]

1991 (1)

1988 (1)

1980 (1)

G. Giuliani and P. Ristori, "Polarization flip cavities: a new approach to laser resonators," Opt. Commun. 35, 109-112 (1980).
[CrossRef]

1971 (1)

W. C. Scott and M. de Wit, "Birefringence compensation and TEM00 mode enhancement in a Nd:YAG laser," Appl. Phys. Lett. 18, 3-4 (1971).
[CrossRef]

1970 (3)

J. D. Foster and L. M. Osterink, "Thermal effects in a Nd:YAG laser," J. Appl. Phys. 41, 3656-3663 (1970).
[CrossRef]

W. Koechner and D. K. Rice, "Effect of birefringence on the performance of linearly polarized YAG:Nd lasers," IEEE J. Quantum Electron. QE-6, 557-566 (1970).
[CrossRef]

W. Koechner, "Thermal lensing in a Nd:YAG laser rod," Appl. Opt. 9, 2548-2553 (1970).
[CrossRef] [PubMed]

Bélanger, P. A.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

de Wit, M.

W. C. Scott and M. de Wit, "Birefringence compensation and TEM00 mode enhancement in a Nd:YAG laser," Appl. Phys. Lett. 18, 3-4 (1971).
[CrossRef]

Dong, S.

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, "A novel approach for compensation of birefringence in cylindrical Nd:YAG rods," Opt. Quantum Electron. 28, 57-69 (1996).
[CrossRef]

Q. Lü, S. Dong, and H. Weber, "Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod," Opt. Quantum Electron. 27, 777-783 (1995).
[CrossRef]

Foster, J. D.

J. D. Foster and L. M. Osterink, "Thermal effects in a Nd:YAG laser," J. Appl. Phys. 41, 3656-3663 (1970).
[CrossRef]

Giuliani, G.

G. Giuliani and P. Ristori, "Polarization flip cavities: a new approach to laser resonators," Opt. Commun. 35, 109-112 (1980).
[CrossRef]

Jackel, S.

I. Moshe, S. Jackel, and A. Meir, "Production of radially or tangentially polarized beams in solid-state lasers and elimination of thermally induced birefringence effects," Opt. Lett. 28, 807-809 (2003).
[CrossRef] [PubMed]

I. Moshe and S. Jackel, "Correction of thermally induced birefringence in double-rod laser resonators--comparison of various methods," Opt. Commun. 214, 315-325 (2002).
[CrossRef]

I. Moshe and S. Jackel, "Correction of birefringence and thermal lensing in nonreciprocal resonators by use of a dynamic imaging mirror," Appl. Opt. 39, 4313-4319 (2000).
[CrossRef]

I. Moshe, S. Jackel, and R. Lallouz, "Working beyond the static limits of laser stability by use of adaptive and polarization-conjugation optics," Appl. Opt. 37, 6415-6420 (1998).
[CrossRef]

S. Jackel, I. Moshe, A. Kaufman, A. Lavi, and R. Lallouz, "High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifier," Opt. Eng. 36, 2031-2036 (1997).
[CrossRef]

I. Moshe, S. Jackel, and A. Meir, "Beam quality improvement in thermally birefringent Nd:YAG laser amplifiers by use of radially polarized beams," in Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2004).

Kaufman, A.

S. Jackel, I. Moshe, A. Kaufman, A. Lavi, and R. Lallouz, "High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifier," Opt. Eng. 36, 2031-2036 (1997).
[CrossRef]

Keren, E.

Koechner, W.

W. Koechner, "Thermal lensing in a Nd:YAG laser rod," Appl. Opt. 9, 2548-2553 (1970).
[CrossRef] [PubMed]

W. Koechner and D. K. Rice, "Effect of birefringence on the performance of linearly polarized YAG:Nd lasers," IEEE J. Quantum Electron. QE-6, 557-566 (1970).
[CrossRef]

W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, New York, 1999).
[CrossRef]

Kugler, N.

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, "A novel approach for compensation of birefringence in cylindrical Nd:YAG rods," Opt. Quantum Electron. 28, 57-69 (1996).
[CrossRef]

Lallouz, R.

I. Moshe, S. Jackel, and R. Lallouz, "Working beyond the static limits of laser stability by use of adaptive and polarization-conjugation optics," Appl. Opt. 37, 6415-6420 (1998).
[CrossRef]

S. Jackel, I. Moshe, A. Kaufman, A. Lavi, and R. Lallouz, "High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifier," Opt. Eng. 36, 2031-2036 (1997).
[CrossRef]

Lavi, A.

S. Jackel, I. Moshe, A. Kaufman, A. Lavi, and R. Lallouz, "High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifier," Opt. Eng. 36, 2031-2036 (1997).
[CrossRef]

Lavi, S.

Lü, Q.

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, "A novel approach for compensation of birefringence in cylindrical Nd:YAG rods," Opt. Quantum Electron. 28, 57-69 (1996).
[CrossRef]

Q. Lü, S. Dong, and H. Weber, "Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod," Opt. Quantum Electron. 27, 777-783 (1995).
[CrossRef]

Meir, A.

I. Moshe, S. Jackel, and A. Meir, "Production of radially or tangentially polarized beams in solid-state lasers and elimination of thermally induced birefringence effects," Opt. Lett. 28, 807-809 (2003).
[CrossRef] [PubMed]

I. Moshe, S. Jackel, and A. Meir, "Beam quality improvement in thermally birefringent Nd:YAG laser amplifiers by use of radially polarized beams," in Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2004).

Moshe, I.

I. Moshe, S. Jackel, and A. Meir, "Production of radially or tangentially polarized beams in solid-state lasers and elimination of thermally induced birefringence effects," Opt. Lett. 28, 807-809 (2003).
[CrossRef] [PubMed]

I. Moshe and S. Jackel, "Correction of thermally induced birefringence in double-rod laser resonators--comparison of various methods," Opt. Commun. 214, 315-325 (2002).
[CrossRef]

I. Moshe and S. Jackel, "Correction of birefringence and thermal lensing in nonreciprocal resonators by use of a dynamic imaging mirror," Appl. Opt. 39, 4313-4319 (2000).
[CrossRef]

I. Moshe, S. Jackel, and R. Lallouz, "Working beyond the static limits of laser stability by use of adaptive and polarization-conjugation optics," Appl. Opt. 37, 6415-6420 (1998).
[CrossRef]

S. Jackel, I. Moshe, A. Kaufman, A. Lavi, and R. Lallouz, "High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifier," Opt. Eng. 36, 2031-2036 (1997).
[CrossRef]

I. Moshe, S. Jackel, and A. Meir, "Beam quality improvement in thermally birefringent Nd:YAG laser amplifiers by use of radially polarized beams," in Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2004).

Müller, N.

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, "A novel approach for compensation of birefringence in cylindrical Nd:YAG rods," Opt. Quantum Electron. 28, 57-69 (1996).
[CrossRef]

Osterink, L. M.

J. D. Foster and L. M. Osterink, "Thermal effects in a Nd:YAG laser," J. Appl. Phys. 41, 3656-3663 (1970).
[CrossRef]

Panofsky, W. K.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, Reading, Mass., 1962).

Phillips, M.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, Reading, Mass., 1962).

Prochaska, R.

Rice, D. K.

W. Koechner and D. K. Rice, "Effect of birefringence on the performance of linearly polarized YAG:Nd lasers," IEEE J. Quantum Electron. QE-6, 557-566 (1970).
[CrossRef]

Ristori, P.

G. Giuliani and P. Ristori, "Polarization flip cavities: a new approach to laser resonators," Opt. Commun. 35, 109-112 (1980).
[CrossRef]

Scott, W. C.

W. C. Scott and M. de Wit, "Birefringence compensation and TEM00 mode enhancement in a Nd:YAG laser," Appl. Phys. Lett. 18, 3-4 (1971).
[CrossRef]

Siegman, A. E.

A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D.A.Holmes, ed., Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N. Y., 1981).

Weber, H.

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, "A novel approach for compensation of birefringence in cylindrical Nd:YAG rods," Opt. Quantum Electron. 28, 57-69 (1996).
[CrossRef]

Q. Lü, S. Dong, and H. Weber, "Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod," Opt. Quantum Electron. 27, 777-783 (1995).
[CrossRef]

Wittrock, U.

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, "A novel approach for compensation of birefringence in cylindrical Nd:YAG rods," Opt. Quantum Electron. 28, 57-69 (1996).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Appl. Opt. (4)

Appl. Phys. Lett. (1)

W. C. Scott and M. de Wit, "Birefringence compensation and TEM00 mode enhancement in a Nd:YAG laser," Appl. Phys. Lett. 18, 3-4 (1971).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. Koechner and D. K. Rice, "Effect of birefringence on the performance of linearly polarized YAG:Nd lasers," IEEE J. Quantum Electron. QE-6, 557-566 (1970).
[CrossRef]

J. Appl. Phys. (1)

J. D. Foster and L. M. Osterink, "Thermal effects in a Nd:YAG laser," J. Appl. Phys. 41, 3656-3663 (1970).
[CrossRef]

Opt. Commun. (2)

G. Giuliani and P. Ristori, "Polarization flip cavities: a new approach to laser resonators," Opt. Commun. 35, 109-112 (1980).
[CrossRef]

I. Moshe and S. Jackel, "Correction of thermally induced birefringence in double-rod laser resonators--comparison of various methods," Opt. Commun. 214, 315-325 (2002).
[CrossRef]

Opt. Eng. (1)

S. Jackel, I. Moshe, A. Kaufman, A. Lavi, and R. Lallouz, "High-energy Nd:Cr:GSGG lasers based on phase and polarization conjugated multiple-pass amplifier," Opt. Eng. 36, 2031-2036 (1997).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (2)

Q. Lü, N. Kugler, H. Weber, S. Dong, N. Müller, and U. Wittrock, "A novel approach for compensation of birefringence in cylindrical Nd:YAG rods," Opt. Quantum Electron. 28, 57-69 (1996).
[CrossRef]

Q. Lü, S. Dong, and H. Weber, "Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod," Opt. Quantum Electron. 27, 777-783 (1995).
[CrossRef]

Other (6)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, New York, 1999).
[CrossRef]

J. T. Verdeyen, Laser Electronics, 2nd ed. (Prentice-Hall, Englewood Cliffs, N. Y., 1981).

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, Reading, Mass., 1962).

A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D.A.Holmes, ed., Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

I. Moshe, S. Jackel, and A. Meir, "Beam quality improvement in thermally birefringent Nd:YAG laser amplifiers by use of radially polarized beams," in Advanced Solid-State Photonics, G.J.Quarles, ed., Vol. 94 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2004).

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Figures (7)

Fig. 1
Fig. 1

Setup to measure birefringence-induced bifocusing.

Fig. 2
Fig. 2

Measured and calculated birefringence-induced beam quality degradation as functions of thermal lensing. Calculations were made using unpolarized incoherent and linearly polarized coherent Gaussian beam models. The probe beams and rod diameters were 3 and 6.35 mm, respectively.

Fig. 3
Fig. 3

Intensity distribution of an unpolarized Gaussian beam along the optical axis after passage through a bifocal lens with focal lengths of f r = 0.30 m and f θ = 0.36 m .

Fig. 4
Fig. 4

Circular polarization. Calculated intensity distribution at the focal planes for the radial and tangential polarized components ( T r , T θ ) and at the CLC plane ( T CLC ) of a bifocal lens with a focal length of of 0.3 m. For comparison, the intensity distribution of a birefringence-free lens is also calculated ( T f ) . The pictures of intensity contours were normalized to their maximal values. To compare intensities, the one-dimensional profiles were normalized to the intensity peak value calculated at the focal plane of the ideal lens. Measurement results of the intensity distribution at the radial and tangential focal planes and at the CLC are also depicted ( M r , M θ , M CLC , respectively).

Fig. 5
Fig. 5

Linear polarization. Calculated intensity distribution at the radial and tangential focal planes and at the CLC ( T r , T θ , T CLC , respectively) of a bifocal lens with a 0.3-m focal length. The solid and dotted curves represent the calculated intensity profile along the X and Y axes, respectively. Experimental results are shown for comparison ( M r , M θ , M CLC ) .

Fig. 6
Fig. 6

Unpolarized beam. Calculated and measured intensity distributions at the focal planes (radial f r , tangential f θ , and at the CLC) of a bifocal lens with a 0.4-m focal length. For comparison, the calculated and measured focal intensity distributions for a birefringence-free lens are shown.

Fig. 7
Fig. 7

Wave-front deformation induced by a bifocal lens on a linearly polarized beam. The thermal focal length was taken as 1 m.

Tables (1)

Tables Icon

Table 1 Optical Beam Amplitude ( A i j _ k ) Values after Propagation through a Thermally Birefringent Rod a

Equations (36)

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

1 q 1 R i λ M 0 2 π ω 2 n .
1 q i = i λ M i 2 π ω i 2 n .
ω r , θ 2 ( z ) = f r , θ 2 π 2 ω i 4 2 π 2 ω i 4 f r , θ z + [ π 2 ω i 4 + ( λ M 0 2 ) 2 f r , θ 2 ] z 2 f r , θ 2 π 2 ω i 2 ,
ω 2 = 4 0 2 π 0 [ r 2 exp ( 2 r 2 ω r 2 ) + r 2 exp ( 2 r 2 ω θ 2 ) ] r d r d θ 0 2 π 0 [ exp ( 2 r 2 ω r 2 ) + exp ( 2 r 2 ω θ 2 ) ] r d r d θ = ω r 4 + ω θ 4 ω r 2 + ω θ 2 .
f CLC = 2 b 1 + b f r ,
f r , θ = 2 n 0 β r , θ l rod ,
β r , θ = η P elect 2 π K l rod R rod 2 ( 1 2 n 0 d n d T + n 0 2 α e C r , θ ) .
C r = ( 17 ν 7 ) P 11 + ( 31 ν 17 ) P 12 + 8 ( ν + 1 ) P 44 48 ( ν 1 ) ,
C θ = ( 10 ν 6 ) P 11 + 2 ( 11 ν 5 ) P 12 32 ( ν 1 ) ,
M 2 = α α G a ω ( f CLC ) ( α G a f CLC ) = ω ( f CLC ) ω G a ( f CLC ) .
ω ( f CLC ) = [ ( 1 b 1 + b ) 2 ω i 2 + ( λ f CLC π ω i ) 2 ] 1 2 .
ω ( f CLC ) = f CLC λ M i 2 π ω i .
M 2 = [ ( M i 2 ) 2 + ( M b 2 ) 2 ] 1 2 ,
M b 2 π ω i 2 λ f CLC ( b 1 b + 1 ) .
ω CLC f CLC 0 = ω i 1 b 1 + b b = 1.2 0.1 ω i .
M 2 = [ 1 + C + ln ( 1 + C ) ] 1 2 ,
C = [ n 0 3 α ( C r C θ ) 2 λ K ω i 2 P h R rod 2 ] 2 ,
M 2 = [ ( M i 2 ) 2 + C + ln ( 1 + C ) ] 1 2 .
I ( z ) = 0.5 I 0 { exp [ 2 r 2 ω r 2 ( z ) ] + exp [ 2 r 2 ω θ 2 ( z ) ] } .
U [ P ( r , ψ , z ) ] = i λ f 2 exp ( i k z ) 0 2 π 0 a U in ( r , ϕ ) exp { i k [ r r f cos ( ϕ ψ ) + 1 2 f 2 z r 2 ] } r d r d ϕ ,
U ( r , ϕ ) i , j _ k = A i , j _ k U in ( r , ϕ ) ,
I ( P ) j = i , k U ( P ) i j _ k 2 .
I ( P ) = j I ( P ) j = I ( P ) X + I ( P ) Y .
U ( P ) Y = U ( P ) X Y _ r + U ( P ) X Y _ θ , U ( P ) X = U ( P ) X X _ r + U ( P ) X X _ θ .
I ( P ) = U ( P ) X 2 + U ( P ) Y 2 .
I ( P ) X = k U ( P ) X X _ k 2 + k U ( P ) Y X _ k 2 ,
I ( P ) Y = k U ( P ) Y Y _ k 2 + k U ( P ) X Y _ k 2 .
I ( P ) = I ( P ) X + I ( P ) Y .
U x x _ r = A x x _ r exp ( i ψ r ) , U x x _ θ = A x x _ θ exp ( i ψ θ ) .
ψ θ = k r 2 2 f θ , ψ r = k r 2 2 f r .
A x x _ θ = A sin 2 ( ϕ ) , A x x _ r = A cos 2 ( ϕ ) .
U x x = U x x _ r + U x x _ θ = A x x exp ( i ψ x x ) ,
A x x = [ A x x _ r 2 + A x x _ θ 2 + 2 A x x _ r A x x _ θ cos ( ψ r ψ θ ) ] 1 2 ,
ψ x x = t g 1 [ A x x _ r sin ( ψ r ) + A x x _ θ sin ( ψ θ ) A x x _ r cos ( ψ r ) + A x x _ θ cos ( ψ θ ) ] .
A y x _ θ = A exp ( i π 2 ) sin ( ϕ ) cos ( ϕ ) ,
A y x _ r = A exp ( i π 2 ) sin ( ϕ ) cos ( ϕ ) .

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