Abstract

We consider second-harmonic generation (SHG) and third-harmonic generation (THG) in a nonlinear optical crystal illuminated by a vector Gaussian beam, i.e., a Gaussian beam in which the axial component of the excitation field is considered. This component exhibits twice the Gouy phase shift of the transverse component and vanishes at points on the beam axis. Harmonic generation stemming from this component exhibits a unique dependence on geometrical factors associated with the location and focusing of the beam relative to the location of the crystal. Using the first Born approximation (undepleted fundamental beam), we derive analytical formulas for the quantities that characterize these geometrical factors for a nonlinear optical crystal described by an arbitrary nonlinear susceptibility tensor, for both SHG and THG and for all polarization components. We also determine the efficiencies of these processes as functions of the geometry of the experimental arrangement for phase-matched crystals as well as for crystals of infinite length.

© 2006 Optical Society of America

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  1. D. A. Kleinman, A. Ashkin, and G. D. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338-379 (1966).
    [CrossRef]
  2. G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3639 (1968).
    [CrossRef]
  3. J. F. Ward and G. H. C. New, "Optical third harmonic generation in gases by a focused laser beam," Phys. Rev. 185, 57-72 (1969).
    [CrossRef]
  4. G. C. Bjorklund, "Effects of focusing on third-order nonlinear processes in isotropic media," IEEE J. Quantum Electron. QE-11, 287-296 (1975).
    [CrossRef]
  5. R. B. Miles and S. E. Harris, "Optical third-harmonic generation in alkali metal vapors," IEEE J. Quantum Electron. QE-9, 470-484 (1973).
    [CrossRef]
  6. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003), Chap. 2.
  7. M. Gouy, "Sur la propagation anomale des ondes," C. R. Hebd. Seances Acad. Sci. 111, 33-40 (1890).
  8. A. E. Siegman, Lasers (University Science, 1986).
  9. R. Gauderon, P. B. Lukins, and C. J. R. Sheppard, "Three-dimensional second-harmonic generation imaging with femtosecond laser pulses," Opt. Lett. 23, 1209-1211 (1998).
    [CrossRef]
  10. P. J. Campagnola, M.-de. Wei, A. Lewis, and L. M. Loew, "High-resolution nonlinear optical imaging of live cells by second harmonic generation," Biophys. J. 77, 3341-3349 (1999).
    [CrossRef] [PubMed]
  11. L. Moreaux, O. Sandre, and J. Mertz, "Membrane imaging by second-harmonic generation microscopy," J. Opt. Soc. Am. B 17, 1685-1694 (2000).
    [CrossRef]
  12. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, "Nonlinear scanning laser microscopy by third harmonic generation," Appl. Phys. Lett. 70, 922-924 (1997).
    [CrossRef]
  13. J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, "Third harmonic generation microscopy," Opt. Express 3, 315-324 (1998).
    [CrossRef] [PubMed]
  14. M. Muller, J. A. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D microscopy of transparent object using third harmonic generation," J. Microsc. 191, 266-274 (1998).
    [CrossRef] [PubMed]
  15. D. Yelin and Y. Silberberg, "Laser scanning third harmonic generation in biology," Opt. Express 5, 169-175 (1999).
    [CrossRef] [PubMed]
  16. D. Oron and Y. Silberberg, "Third-harmonic generation with cylindrical Gaussian beams," J. Opt. Soc. Am. B 21, 1964-1968 (2004).
    [CrossRef]
  17. D. Debarre, W. Supatto, and E. Beaurepatre, "Structure sensitivity in third-harmonic generation microscopy," Opt. Lett. 30, 2134-2136 (2005).
    [CrossRef] [PubMed]
  18. J.-X. Cheng and X. S. Xie, "Green's function formulation for third-harmonic generation microscopy," J. Opt. Soc. Am. B 19, 1604-1610 (2002).
    [CrossRef]
  19. E. Yew and C. Sheppard, "Effects of axial field components on second harmonic generation microscopy," Opt. Express 14, 1167-1174 (2006).
    [CrossRef] [PubMed]
  20. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  21. R. W. Hellwarth, "Third-order optical susceptibilities of liquids and solids," Prog. Quantum Electron. 5, 1-68 (1977).
    [CrossRef]
  22. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chaps. 3 and 19.
    [CrossRef]
  23. A. F. Abouraddy and K. C. Toussaint, Jr., "Three-dimensional polarization control in microscopy," Phys. Rev. Lett. 96, 153901 (2006).
    [CrossRef] [PubMed]

2006 (2)

A. F. Abouraddy and K. C. Toussaint, Jr., "Three-dimensional polarization control in microscopy," Phys. Rev. Lett. 96, 153901 (2006).
[CrossRef] [PubMed]

E. Yew and C. Sheppard, "Effects of axial field components on second harmonic generation microscopy," Opt. Express 14, 1167-1174 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

2002 (1)

2000 (1)

1999 (2)

D. Yelin and Y. Silberberg, "Laser scanning third harmonic generation in biology," Opt. Express 5, 169-175 (1999).
[CrossRef] [PubMed]

P. J. Campagnola, M.-de. Wei, A. Lewis, and L. M. Loew, "High-resolution nonlinear optical imaging of live cells by second harmonic generation," Biophys. J. 77, 3341-3349 (1999).
[CrossRef] [PubMed]

1998 (3)

1997 (1)

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, "Nonlinear scanning laser microscopy by third harmonic generation," Appl. Phys. Lett. 70, 922-924 (1997).
[CrossRef]

1977 (1)

R. W. Hellwarth, "Third-order optical susceptibilities of liquids and solids," Prog. Quantum Electron. 5, 1-68 (1977).
[CrossRef]

1975 (1)

G. C. Bjorklund, "Effects of focusing on third-order nonlinear processes in isotropic media," IEEE J. Quantum Electron. QE-11, 287-296 (1975).
[CrossRef]

1973 (1)

R. B. Miles and S. E. Harris, "Optical third-harmonic generation in alkali metal vapors," IEEE J. Quantum Electron. QE-9, 470-484 (1973).
[CrossRef]

1969 (1)

J. F. Ward and G. H. C. New, "Optical third harmonic generation in gases by a focused laser beam," Phys. Rev. 185, 57-72 (1969).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3639 (1968).
[CrossRef]

1966 (1)

D. A. Kleinman, A. Ashkin, and G. D. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338-379 (1966).
[CrossRef]

1890 (1)

M. Gouy, "Sur la propagation anomale des ondes," C. R. Hebd. Seances Acad. Sci. 111, 33-40 (1890).

Abouraddy, A. F.

A. F. Abouraddy and K. C. Toussaint, Jr., "Three-dimensional polarization control in microscopy," Phys. Rev. Lett. 96, 153901 (2006).
[CrossRef] [PubMed]

Ashkin, A.

D. A. Kleinman, A. Ashkin, and G. D. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338-379 (1966).
[CrossRef]

Barad, Y.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, "Nonlinear scanning laser microscopy by third harmonic generation," Appl. Phys. Lett. 70, 922-924 (1997).
[CrossRef]

Beaurepatre, E.

Bjorklund, G. C.

G. C. Bjorklund, "Effects of focusing on third-order nonlinear processes in isotropic media," IEEE J. Quantum Electron. QE-11, 287-296 (1975).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3639 (1968).
[CrossRef]

D. A. Kleinman, A. Ashkin, and G. D. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338-379 (1966).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003), Chap. 2.

Brakenhoff, G. J.

M. Muller, J. A. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D microscopy of transparent object using third harmonic generation," J. Microsc. 191, 266-274 (1998).
[CrossRef] [PubMed]

J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, "Third harmonic generation microscopy," Opt. Express 3, 315-324 (1998).
[CrossRef] [PubMed]

Campagnola, P. J.

P. J. Campagnola, M.-de. Wei, A. Lewis, and L. M. Loew, "High-resolution nonlinear optical imaging of live cells by second harmonic generation," Biophys. J. 77, 3341-3349 (1999).
[CrossRef] [PubMed]

Cheng, J.-X.

Debarre, D.

Eisenberg, H.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, "Nonlinear scanning laser microscopy by third harmonic generation," Appl. Phys. Lett. 70, 922-924 (1997).
[CrossRef]

Gauderon, R.

Gouy, M.

M. Gouy, "Sur la propagation anomale des ondes," C. R. Hebd. Seances Acad. Sci. 111, 33-40 (1890).

Harris, S. E.

R. B. Miles and S. E. Harris, "Optical third-harmonic generation in alkali metal vapors," IEEE J. Quantum Electron. QE-9, 470-484 (1973).
[CrossRef]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

Hellwarth, R. W.

R. W. Hellwarth, "Third-order optical susceptibilities of liquids and solids," Prog. Quantum Electron. 5, 1-68 (1977).
[CrossRef]

Horowitz, M.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, "Nonlinear scanning laser microscopy by third harmonic generation," Appl. Phys. Lett. 70, 922-924 (1997).
[CrossRef]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3639 (1968).
[CrossRef]

D. A. Kleinman, A. Ashkin, and G. D. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338-379 (1966).
[CrossRef]

Lewis, A.

P. J. Campagnola, M.-de. Wei, A. Lewis, and L. M. Loew, "High-resolution nonlinear optical imaging of live cells by second harmonic generation," Biophys. J. 77, 3341-3349 (1999).
[CrossRef] [PubMed]

Loew, L. M.

P. J. Campagnola, M.-de. Wei, A. Lewis, and L. M. Loew, "High-resolution nonlinear optical imaging of live cells by second harmonic generation," Biophys. J. 77, 3341-3349 (1999).
[CrossRef] [PubMed]

Lukins, P. B.

Mertz, J.

Miles, R. B.

R. B. Miles and S. E. Harris, "Optical third-harmonic generation in alkali metal vapors," IEEE J. Quantum Electron. QE-9, 470-484 (1973).
[CrossRef]

Moreaux, L.

Muller, M.

J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, "Third harmonic generation microscopy," Opt. Express 3, 315-324 (1998).
[CrossRef] [PubMed]

M. Muller, J. A. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D microscopy of transparent object using third harmonic generation," J. Microsc. 191, 266-274 (1998).
[CrossRef] [PubMed]

New, G. H. C.

J. F. Ward and G. H. C. New, "Optical third harmonic generation in gases by a focused laser beam," Phys. Rev. 185, 57-72 (1969).
[CrossRef]

Oron, D.

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chaps. 3 and 19.
[CrossRef]

Sandre, O.

Sheppard, C.

Sheppard, C. J. R.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Silberberg, Y.

Squier, J. A.

J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, "Third harmonic generation microscopy," Opt. Express 3, 315-324 (1998).
[CrossRef] [PubMed]

M. Muller, J. A. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D microscopy of transparent object using third harmonic generation," J. Microsc. 191, 266-274 (1998).
[CrossRef] [PubMed]

Supatto, W.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chaps. 3 and 19.
[CrossRef]

Toussaint, K. C.

A. F. Abouraddy and K. C. Toussaint, Jr., "Three-dimensional polarization control in microscopy," Phys. Rev. Lett. 96, 153901 (2006).
[CrossRef] [PubMed]

Ward, J. F.

J. F. Ward and G. H. C. New, "Optical third harmonic generation in gases by a focused laser beam," Phys. Rev. 185, 57-72 (1969).
[CrossRef]

Wei, M.-de.

P. J. Campagnola, M.-de. Wei, A. Lewis, and L. M. Loew, "High-resolution nonlinear optical imaging of live cells by second harmonic generation," Biophys. J. 77, 3341-3349 (1999).
[CrossRef] [PubMed]

Wilson, K. R.

M. Muller, J. A. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D microscopy of transparent object using third harmonic generation," J. Microsc. 191, 266-274 (1998).
[CrossRef] [PubMed]

J. A. Squier, M. Muller, G. J. Brakenhoff, and K. R. Wilson, "Third harmonic generation microscopy," Opt. Express 3, 315-324 (1998).
[CrossRef] [PubMed]

Xie, X. S.

Yelin, D.

Yew, E.

Appl. Phys. Lett. (1)

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, "Nonlinear scanning laser microscopy by third harmonic generation," Appl. Phys. Lett. 70, 922-924 (1997).
[CrossRef]

Biophys. J. (1)

P. J. Campagnola, M.-de. Wei, A. Lewis, and L. M. Loew, "High-resolution nonlinear optical imaging of live cells by second harmonic generation," Biophys. J. 77, 3341-3349 (1999).
[CrossRef] [PubMed]

C. R. Hebd. Seances Acad. Sci. (1)

M. Gouy, "Sur la propagation anomale des ondes," C. R. Hebd. Seances Acad. Sci. 111, 33-40 (1890).

IEEE J. Quantum Electron. (2)

G. C. Bjorklund, "Effects of focusing on third-order nonlinear processes in isotropic media," IEEE J. Quantum Electron. QE-11, 287-296 (1975).
[CrossRef]

R. B. Miles and S. E. Harris, "Optical third-harmonic generation in alkali metal vapors," IEEE J. Quantum Electron. QE-9, 470-484 (1973).
[CrossRef]

J. Appl. Phys. (1)

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3639 (1968).
[CrossRef]

J. Microsc. (1)

M. Muller, J. A. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D microscopy of transparent object using third harmonic generation," J. Microsc. 191, 266-274 (1998).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (3)

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. (2)

J. F. Ward and G. H. C. New, "Optical third harmonic generation in gases by a focused laser beam," Phys. Rev. 185, 57-72 (1969).
[CrossRef]

D. A. Kleinman, A. Ashkin, and G. D. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338-379 (1966).
[CrossRef]

Phys. Rev. Lett. (1)

A. F. Abouraddy and K. C. Toussaint, Jr., "Three-dimensional polarization control in microscopy," Phys. Rev. Lett. 96, 153901 (2006).
[CrossRef] [PubMed]

Prog. Quantum Electron. (1)

R. W. Hellwarth, "Third-order optical susceptibilities of liquids and solids," Prog. Quantum Electron. 5, 1-68 (1977).
[CrossRef]

Other (4)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), Chaps. 3 and 19.
[CrossRef]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

A. E. Siegman, Lasers (University Science, 1986).

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003), Chap. 2.

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Figures (4)

Fig. 1
Fig. 1

Geometries for harmonic generation: (a) centered crystal, (b) offset crystal.

Fig. 2
Fig. 2

(a) Optical intensities of second-harmonic components at z = 10 z 0 for an offset crystal with L = 2 z 0 , i.e., ξ = 1 . The fundamental wave is assumed to be linearly polarized in the x direction, so that α = 1 and β = 0 . The angle of divergence of the fundamental beam is θ 0 = 67 mrad = 3.8 ° . The x x component is Gaussian, the x z component is Hermite–Gaussian of order (1,0), and the z z component is a superposition of Hermite–Gaussian functions of orders (0,0) and (2,0). (b) Optical intensities of third-harmonic components at z = 10 z 0 for an offset crystal with L = 2 z 0 , i.e., ξ = 1 . The fundamental wave is assumed to be linearly polarized in the x direction, so that α = 1 and β = 0 . The angle of divergence of the fundamental beam is θ 0 = 67 mrad = 3.8 ° . The x x x component is Gaussian, the x x z component is Hermite–Gaussian of order (1,0), and the x z z component is a superposition of Hermite–Gaussian functions of order (0,0) and (2,0). The z z z component is a sum of Hermite–Gaussian functions of order (1,0) and (3,0).

Fig. 3
Fig. 3

Dependence of the magnitudes of the efficiency factors η 1 ( m ) and η 2 ( m ) on the geometrical factor ξ = L 2 z 0 for SHG ( m = 2 ) and THG ( m = 3 ) in the phase-matched case with a centered crystal (solid curves) and with an offset crystal (dashed curves).

Fig. 4
Fig. 4

Dependence of the magnitudes of the efficiency factors η 1 ( m ) and η 2 ( m ) on the phase mismatch factor χ = Δ k z 0 for SHG ( m = 2 ) and THG ( m = 3 ) assuming a long crystal ( ξ = L 2 z 0 = 10 3 ) that is centered (solid curves) or offset (dashed curves).

Equations (52)

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G ( ρ , z , k ) = j z 0 z + j z 0 exp ( j k ρ 2 2 ( z + j z 0 ) ) exp ( j k z ) ,
E ( ω ) ( ρ , z ) = G ( ρ , z , k ) [ α x ̂ β y ̂ + α x + β y z + j z 0 z ̂ ] ,
E x ω = α G ( ρ , z , k ) ,
E y ω = β G ( ρ , z , k ) ,
E z ω = G ( ρ , z , k ) α x + β y z + j z 0 .
P i ( 2 ) = j k d i j k ( 2 ) E j ω E k ω , i , j , k = x , y , z ,
P i ( 3 ) = j k l χ i j k l ( 3 ) E j ω E k ω E l ω , i , j , k , l = x , y , z ,
S i ( 2 ) = μ o ω 2 j k d i j k ( 2 ) E j ω E k ω ,
S i ( 3 ) = μ o ω 2 j k l χ i j k l ( 3 ) E j ω E k ω E l ω ,
S i ( 2 ) = μ o ω 2 [ d i x x ( 2 ) S x x + d i y y ( 2 ) S y y + d i x y ( 2 ) S x y + d i z z ( 2 ) S z z + d i x z ( 2 ) S x z + d i y z ( 2 ) S y z ] ,
S i ( 3 ) = μ o ω 2 [ χ i x x x ( 3 ) S x x x + χ i y y y ( 3 ) S y y y + χ i x y y ( 3 ) S x y y + χ i y x x ( 3 ) S y x x + χ i z z z ( 3 ) S z z z + χ i x z z ( 3 ) S x z z + χ i y z z ( 3 ) S y z z + χ i x x z ( 3 ) S x x z + χ i y y z ( 3 ) S y y z + χ i x y z ( 3 ) S x y z ] ,
S x x = E x ω E x ω , S y y = E y ω E y ω , S z z = E z ω E z ω , S x y = 2 E x ω E y ω , S x z = 2 E x ω E z ω , S y z = 2 E y ω E z ω ,
S x x x = E x ω E x ω E x ω , S y y y = E y ω E y ω E y ω , S z z z = E z ω E z ω E z ω ,
S x y y = 3 E x ω E y ω E y ω , S y x x = 3 E x ω E x ω E y ω , S x z z = 3 E x ω E z ω E z ω ,
S y z z = 3 E y ω E z ω E z ω , S z x x = 3 E x ω E x ω E z ω , S z y y = 3 E y ω E y ω E z ω ,
S x y z = 3 E x ω E y ω E z ω .
χ y y z z ( 3 ) = χ z z y y ( 3 ) = χ z z x x ( 3 ) = χ x x z z ( 3 ) = χ x x y y ( 3 ) = χ y y x x ( 3 ) ,
χ y z y z ( 3 ) = χ z y z y ( 3 ) = χ z x z x ( 3 ) = χ x z x z ( 3 ) = χ x y x y ( 3 ) = χ y x y x ( 3 ) ,
χ y z z y ( 3 ) = χ z y y z ( 3 ) = χ z x x z ( 3 ) = χ x x z z ( 3 ) = χ x y y x ( 3 ) = χ y x x y ( 3 ) ,
χ x x x x ( 3 ) = χ y y y y ( 3 ) = χ z z z z ( 3 ) = χ x x y y ( 3 ) + χ x y x y ( 3 ) + χ x y y x ( 3 ) .
S x ( 3 ) = μ o ω 2 χ x x x x ( 3 ) [ S x x x + 1 3 S x y y + 1 3 S x z z ] ,
S y ( 3 ) = μ o ω 2 χ y y y y ( 3 ) [ S y y y + 1 3 S y x x + 1 3 S y z z ] ,
S z ( 3 ) = μ o ω 2 χ z z z z ( 3 ) [ S z z z + 1 3 S z x x + 1 3 S z y y ] .
E ( m ω ) ( x , y , z , z ) h 0 ( z z ) S ( x , y , z ) exp ( j k m ( x x ) 2 + ( y y ) 2 2 ( z z ) ) d x d y ,
E ( m ω ) ( x , y , z ) = crystal E ( m ω ) ( x , y , z , z ) d z ,
E x x 2 ω α 2 η 1 ( 2 ) G ( ρ , z , k 2 ) ,
E y y 2 ω β 2 η 1 ( 2 ) G ( ρ , z , k 2 ) ,
E x y 2 ω 2 α β η 1 ( 2 ) G ( ρ , z , k 2 ) .
E x z 2 ω 2 α η 1 ( 2 ) G ( ρ , z , k 2 ) α x + β y z + j z 0 ,
E y z 2 ω 2 β η 1 ( 2 ) G ( ρ , z , k 2 ) α x + β y z + j z 0 .
E z z 2 ω G ( ρ , z , k 2 ) [ η 1 ( 2 ) ( α x + β y z + j z 0 ) 2 ( α 2 + β 2 ) θ 0 2 4 ( η 2 ( 2 ) η 1 ( 2 ) j z 0 z + j z 0 ) ] ,
η 1 ( 2 ) = j z 1 z 2 d z e j Δ k z z + j z 0 , η 2 ( 2 ) = z 0 z 1 z 2 d z e j Δ k z ( z + j z 0 ) 2
E x x x 3 ω α 3 η 1 ( 3 ) G ( ρ , z , k 3 ) ,
E y y y 3 ω β 3 η 1 ( 3 ) G ( ρ , z , k 3 ) ,
E x y y 3 ω 3 α β 2 η 1 ( 3 ) G ( ρ , z , k 3 ) ,
E x x y 3 ω 3 α 2 β η 1 ( 3 ) G ( ρ , z , k 3 ) .
E x x z 3 ω 3 α 2 η 1 ( 3 ) G ( ρ , z , k 3 ) α x + β y z + j z 0 ,
E y y z 3 ω 3 β 2 η 1 ( 3 ) G ( ρ , z , k 3 ) α x + β y z + j z 0 ,
E x y z 3 ω 3 α β η 1 ( 3 ) G ( ρ , z , k 3 ) α x + β y z + j z 0 .
E x z z 3 ω 3 α G ( ρ , z , k 3 ) [ η 1 ( 3 ) ( α x + β y z + j z 0 ) 2 ( α 2 + β 2 ) θ 0 2 6 ( η 2 ( 3 ) η 1 ( 3 ) j z 0 z + j z 0 ) ] ,
E y z z 3 ω 3 β G ( ρ , z , k 3 ) [ η 1 ( 3 ) ( α x + β y z + j z 0 ) 2 ( α 2 + β 2 ) θ 0 2 6 ( η 2 ( 3 ) η 1 ( 3 ) j z 0 z + j z 0 ) ] .
E z z z 3 ω G ( ρ , z , k 3 ) { η 1 ( 3 ) ( α x + β y z + j z 0 ) 3 ( α 2 + β 2 ) θ 0 2 2 ( η 2 ( 3 ) η 1 ( 3 ) j z 0 z + j z 0 ) ( α x + β y z + j z 0 ) } .
η 1 ( 3 ) = z 0 z 1 z 2 d z e j Δ k z ( z + j z 0 ) 2 , η 2 ( 3 ) = j z 0 2 z 1 z 2 d z e j Δ k z ( z + j z 0 ) 3 .
η 1 ( m ) = ( j z 0 ) m 2 z 1 z 2 d z j e j Δ k z ( z + j z 0 ) m 1 ,
η 2 ( m ) = ( j z 0 ) m 1 z 1 z 2 d z j e j Δ k z ( z + j z 0 ) m .
η 1 ( 2 ) = 2 tan 1 ξ , η 2 ( 2 ) = 2 ξ 1 + ξ 2 ,
η 1 ( 2 ) = j ln ( 1 j 2 ξ ) , η 2 ( 2 ) = 2 ξ 1 j 2 ξ ,
η 1 ( 2 ) = 2 π e χ V ( χ ) , η 2 ( 2 ) = 2 π χ e χ U ( χ ) ,
U ( χ ) = { 0 , χ 0 1 , χ > 0 } , V ( χ ) = { 0 , χ < 0 1 2 , χ = 0 1 , χ > 0 } .
η 1 ( 3 ) = 2 ξ 1 + ξ 2 , η 2 ( 3 ) = 2 ξ ( 1 + ξ 2 ) 2 .
η 1 ( 3 ) = 2 ξ 1 j 2 ξ , η 2 ( 3 ) = 2 ξ ( 1 j ξ ) ( 1 j 2 ξ ) 2 .
η 1 ( 3 ) = 2 π χ e χ U ( χ ) , η 2 ( 3 ) = π χ 2 e χ U ( χ ) .

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