Abstract

We present a new, to the best of our knowledge, formalism based on the propagation of variances in an ABCD optical system to study nonlinear effect in a Kerr medium. This theory, developed for the first order in the beam power, is applied to modal laser fields (Gaussian, Hermite–Gaussian, and Laguerre–Gaussian beams) and permits one to obtain a simple analytical formulation for self-focusing and Z-scan experiments, whatever the thickness of the nonlinear medium.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. K. Buse and M. Luennemann, "3D imaging: wave front sensing utilizing a birefringent crystal," Phys. Rev. Lett. 85, 3385-3387 (2000).
    [CrossRef] [PubMed]
  4. W.Mecklenbräuker and F.Hlawatsch, eds., The Wigner Distribution: Theory and Application in Signal Processing (Elsevier, 1997).
  5. D. Dragoman, "The Wigner distribution function in optics and optoelectronics," Progress in Optics XXXVII, E.Wolf, ed. (Elseiver Science, 1997) pp. 3-53.
  6. A.E.Siegman, ed. Lasers (University Science, 1986).
  7. F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
    [CrossRef]
  8. 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]
  9. M. J. Bastiaans, "Second-order moments of the Wigner distribution function in first-order optical systems," Optik (Stuttgart) 88, 163-168 (1991).
  10. M. J. Bastiaans, "ABCD law for partially coherent Gaussian light, propagation through first-order optical systems," Opt. Quantum Electron. 24, 1011-1019 (1992).
    [CrossRef]
  11. M. J. Bastiaans, "Wigner distribution function applied to twisted Gaussian light propagation in first-order optical systems," J. Opt. Soc. Am. A 17, 2475-2480 (2000).
    [CrossRef]
  12. M. J. Bastiaans, "The Wigner distribution function and its application to first order optics," J. Opt. Soc. Am. 69, 1710-1716 (1979).
    [CrossRef]
  13. M. J. Bastiaans, "Application of the Wigner distribution function in optics," http://www.sps.ele.tue.nl/members/M.J.Bastiaans/abstracts/wigner.html.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  22. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
    [CrossRef]
  23. M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, "High-sensitivity single-beam n2 measurement," Opt. Lett. 14, 955-957 (1989).
    [CrossRef] [PubMed]
  24. M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. (Bellingham) 30, 1228-1235 (1991).
    [CrossRef]
  25. J. A. Hermann and R. G. McDuff, "Analysis of spatial scanning with thick optically nonlinear media," J. Opt. Soc. Am. B 10, 2056-2064 (1993).
    [CrossRef]
  26. J. G. Tian, W. P. Zang, C. Z. Zhang, and G. Zhang, "Analysis of beam propagation in thick nonlinear media," Appl. Opt. 34, 4331-4336 (1995).
    [CrossRef] [PubMed]

2000

K. Buse and M. Luennemann, "3D imaging: wave front sensing utilizing a birefringent crystal," Phys. Rev. Lett. 85, 3385-3387 (2000).
[CrossRef] [PubMed]

M. J. Bastiaans, "Wigner distribution function applied to twisted Gaussian light propagation in first-order optical systems," J. Opt. Soc. Am. A 17, 2475-2480 (2000).
[CrossRef]

1996

D. Dragoman, "Wigner distribution function in nonlinear optics," Appl. Opt. 35, 4142-4146 (1996).
[CrossRef] [PubMed]

A. E. Siegman, "Modes, beams, coherence and orthogonality," Proc. SPIE 2870, 250-259 (1996).
[CrossRef]

1995

1993

1992

C. Paré and P. A. Bélanger, "Beam propagation in a linear of nonlinear lens-like medium using ABCD ray matrices: the method of moments," Opt. Quantum Electron. 24, 1051-1070 (1992).
[CrossRef]

M. J. Bastiaans, "ABCD law for partially coherent Gaussian light, propagation through first-order optical systems," Opt. Quantum Electron. 24, 1011-1019 (1992).
[CrossRef]

D. Huang, M. Ulman, L. H. Acioli, H. A. Haus, and J. G. Fujimoto, "Self-focusing-induced saturable loss for laser mode locking," Opt. Lett. 17, 511-513 (1992).
[CrossRef] [PubMed]

1991

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. (Bellingham) 30, 1228-1235 (1991).
[CrossRef]

R. Martinez-Herrero and P. M. Mejias, "Beam characterization through active media," Opt. Commun. 85, 162-166 (1991).
[CrossRef]

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

M. J. Bastiaans, "Second-order moments of the Wigner distribution function in first-order optical systems," Optik (Stuttgart) 88, 163-168 (1991).

F. Salin, J. Squier, and M. Piché, "Mode locking of Ti:Al2O3 lasers and self-focusing: a Gaussian approximation," Opt. Lett. 16, 1674-1676 (1991).
[CrossRef] [PubMed]

1990

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

1989

1988

1979

Acioli, L. H.

Alda, J.

Bastiaans, M. J.

M. J. Bastiaans, "Wigner distribution function applied to twisted Gaussian light propagation in first-order optical systems," J. Opt. Soc. Am. A 17, 2475-2480 (2000).
[CrossRef]

M. J. Bastiaans, "ABCD law for partially coherent Gaussian light, propagation through first-order optical systems," Opt. Quantum Electron. 24, 1011-1019 (1992).
[CrossRef]

M. J. Bastiaans, "Second-order moments of the Wigner distribution function in first-order optical systems," Optik (Stuttgart) 88, 163-168 (1991).

M. J. Bastiaans, "The Wigner distribution function and its application to first order optics," J. Opt. Soc. Am. 69, 1710-1716 (1979).
[CrossRef]

M. J. Bastiaans, "Application of the Wigner distribution function in optics," http://www.sps.ele.tue.nl/members/M.J.Bastiaans/abstracts/wigner.html.

Bélanger, P. A.

C. Paré and P. A. Bélanger, "Beam propagation in a linear of nonlinear lens-like medium using ABCD ray matrices: the method of moments," Opt. Quantum Electron. 24, 1051-1070 (1992).
[CrossRef]

Bernabeu, E.

Buse, K.

K. Buse and M. Luennemann, "3D imaging: wave front sensing utilizing a birefringent crystal," Phys. Rev. Lett. 85, 3385-3387 (2000).
[CrossRef] [PubMed]

Cerullo, G.

V. Magni, G. Cerullo, and S. DeSilvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348-355 (1993).
[CrossRef]

DeSilvestri, S.

V. Magni, G. Cerullo, and S. DeSilvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348-355 (1993).
[CrossRef]

Dragoman, D.

Fujimoto, J. G.

Gori, F.

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. (Bellingham) 30, 1228-1235 (1991).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Haus, H. A.

Hermann, J. A.

Huang, D.

Keren, E.

Lavi, S.

Luennemann, M.

K. Buse and M. Luennemann, "3D imaging: wave front sensing utilizing a birefringent crystal," Phys. Rev. Lett. 85, 3385-3387 (2000).
[CrossRef] [PubMed]

Magni, V.

V. Magni, G. Cerullo, and S. DeSilvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348-355 (1993).
[CrossRef]

Martinez-Herrero, R.

R. Martinez-Herrero and P. M. Mejias, "Beam characterization through active media," Opt. Commun. 85, 162-166 (1991).
[CrossRef]

McDuff, R. G.

Mejias, P. M.

R. Martinez-Herrero and P. M. Mejias, "Beam characterization through active media," Opt. Commun. 85, 162-166 (1991).
[CrossRef]

Paré, C.

C. Paré and P. A. Bélanger, "Beam propagation in a linear of nonlinear lens-like medium using ABCD ray matrices: the method of moments," Opt. Quantum Electron. 24, 1051-1070 (1992).
[CrossRef]

Piché, M.

Porras, M. A.

Primot, J.

Prochaska, R.

Said, A. A.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. (Bellingham) 30, 1228-1235 (1991).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, "High-sensitivity single-beam n2 measurement," Opt. Lett. 14, 955-957 (1989).
[CrossRef] [PubMed]

Salin, F.

Santarsiero, M.

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. (Bellingham) 30, 1228-1235 (1991).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, "High-sensitivity single-beam n2 measurement," Opt. Lett. 14, 955-957 (1989).
[CrossRef] [PubMed]

Siegman, A. E.

A. E. Siegman, "Modes, beams, coherence and orthogonality," Proc. SPIE 2870, 250-259 (1996).
[CrossRef]

Soileau, M. J.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. (Bellingham) 30, 1228-1235 (1991).
[CrossRef]

Squier, J.

Tian, J. G.

Ulman, M.

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. (Bellingham) 30, 1228-1235 (1991).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, "High-sensitivity single-beam n2 measurement," Opt. Lett. 14, 955-957 (1989).
[CrossRef] [PubMed]

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Zang, W. P.

Zhang, C. Z.

Zhang, G.

Appl. Opt.

IEEE J. Quantum Electron.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

V. Magni, G. Cerullo, and S. DeSilvestri, "ABCD matrix analysis of propagation of Gaussian beams through Kerr media," Opt. Commun. 96, 348-355 (1993).
[CrossRef]

R. Martinez-Herrero and P. M. Mejias, "Beam characterization through active media," Opt. Commun. 85, 162-166 (1991).
[CrossRef]

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

Opt. Eng. (Bellingham)

M. Sheik-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, and E. W. Van Stryland, "Nonlinear refraction and optical limiting in thick media," Opt. Eng. (Bellingham) 30, 1228-1235 (1991).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

C. Paré and P. A. Bélanger, "Beam propagation in a linear of nonlinear lens-like medium using ABCD ray matrices: the method of moments," Opt. Quantum Electron. 24, 1051-1070 (1992).
[CrossRef]

M. J. Bastiaans, "ABCD law for partially coherent Gaussian light, propagation through first-order optical systems," Opt. Quantum Electron. 24, 1011-1019 (1992).
[CrossRef]

Optik (Stuttgart)

M. J. Bastiaans, "Second-order moments of the Wigner distribution function in first-order optical systems," Optik (Stuttgart) 88, 163-168 (1991).

Phys. Rev. Lett.

K. Buse and M. Luennemann, "3D imaging: wave front sensing utilizing a birefringent crystal," Phys. Rev. Lett. 85, 3385-3387 (2000).
[CrossRef] [PubMed]

Proc. SPIE

A. E. Siegman, "Modes, beams, coherence and orthogonality," Proc. SPIE 2870, 250-259 (1996).
[CrossRef]

Other

W.Mecklenbräuker and F.Hlawatsch, eds., The Wigner Distribution: Theory and Application in Signal Processing (Elsevier, 1997).

D. Dragoman, "The Wigner distribution function in optics and optoelectronics," Progress in Optics XXXVII, E.Wolf, ed. (Elseiver Science, 1997) pp. 3-53.

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

M. J. Bastiaans, "Application of the Wigner distribution function in optics," http://www.sps.ele.tue.nl/members/M.J.Bastiaans/abstracts/wigner.html.

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Figures (3)

Fig. 1
Fig. 1

Graphical representation of the vector u according to the coordinate system.

Fig. 2
Fig. 2

Schematic defining parameters z and z m .

Fig. 3
Fig. 3

Z-scan variance versus the position z ¯ m of the sample. (a) For a thin sample e ¯ n 0 = 0.1 with P = 0.1 P N L according to Eq. (39) (solid curves). For a thin sample we can also use Eq. (35) for a Gaussian beam (solid circles). Notice that the condition P = 0.1 P N L corresponds to n 2 I 0 = 0.1 w 0 2 ( n 0 z R 2 ) , where I 0 is the maximum irradiance at focus for a Gaussian beam. (b) For a thick sample e ¯ n 0 = 20 with P=0.1 P N L according to Eq. (39).

Tables (2)

Tables Icon

Table 1 Variance Values and Rayleigh Length According to the Field Shape a

Tables Icon

Table 2 Nonlinear Power P N L According to the Beam Shape of Modal Fields

Equations (60)

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

σ r u 2 = ( r u ) 2 I r u I 2 ,
σ r u 2 = ( r u r u I ) 2 W ( r , k ) d 2 r d 2 k W ( r , k ) d 2 r d 2 k ,
W ( r , k ) = E ( r + s 2 ) E ( r s 2 ) * exp ( i k s ) d 2 s .
σ k u 2 = ( k u ) 2 I ̃ k u I ̃ 2 ,
[ r u k u k 0 ] out = [ A B C D ] [ r u k u k 0 ] in ,
W out ( r , k ) = W in ( D r B k k 0 , C k 0 r + A k ) .
[ σ r u 2 σ k u 2 ] out = [ A 2 B 2 k 0 2 C 2 k 0 2 D 2 ] [ σ r u 2 σ k u 2 ] waist ,
[ σ x 2 σ k x 2 ] out = [ A 2 B 2 k 0 2 C 2 k 0 2 D 2 ] [ σ x 2 σ k x 2 ] waist ,
[ σ y 2 σ k y 2 ] out = [ A 2 B 2 k 0 2 C 2 k 0 2 D 2 ] [ σ y 2 σ k y 2 ] waist .
σ r 2 = r 2 I ( r ) d 2 r I ( r ) d 2 r = σ x 2 + σ y 2 ,
σ k 2 = k 2 I ̃ ( k ) d 2 k I ̃ ( k ) d 2 k = σ k x 2 + σ k y 2 ,
σ r 2 = ( A 2 + B 2 z R 2 ) σ r , waist 2 ,
σ k 2 = ( C 2 z R 2 + D 2 ) σ k , waist 2 ,
σ r , L ( z ) 2 = σ r , waist 2 ( 1 + z ¯ 2 ) σ k , L ( z ) 2 = σ k , waist 2 with M L = [ 1 z 0 1 ] ,
E N L ( r , ε ) = E ( r , z m ) exp [ i φ N L ( r , z m ) ] , φ N L ( r , z m ) = k 0 ε n 2 I ( r , z m ) ,
σ r u ( ε ) 2 = ( r u ) 2 I N L ( r , ε ) d 2 r I N L ( r , ε ) d 2 r = σ r u , L ( z m ) 2 = σ r u , waist 2 ( 1 + z ¯ m 2 ) ,
σ k u ( ε ) 2 = ( u k ) 2 I ̃ N L ( k , ε ) d 2 k I ̃ N L ( k , ε ) d 2 k , = [ u k E ̃ N L ( k , ε ) ] [ u k E ̃ N L ( k , ε ) ] * d 2 k E ̃ N L ( k , ε ) 2 d 2 k ,
σ k u ( ε ) 2 = [ ( u ) E N L ( r , ε ) ] [ ( u ) E N L ( r , ε ) ] * d 2 r E N L ( r , ε ) 2 d 2 r .
σ k u ( ε ) 2 = σ k u , L ( z m ) 2 + 2 [ ( u ) φ N L ( r , z m ) ] [ ( u ) φ ( r , z m ) ] I ( r , z m ) d 2 r I ( r , z m ) d 2 r ,
σ k u ( ε ) 2 σ k u , L ( z m ) 2 = P P N L ( σ r , waist 2 σ r , L ( z m ) 2 ) 2 ε ¯ n 0 z ¯ m σ k , waist 2 ,
P N L = 1 α λ 2 2 π n 0 n 2 .
α = 2 π σ k , waist 2 I ( r , z m = 0 ) 2 d 2 r [ I ( r , z m = 0 ) d 2 r ] 2 ,
σ r ( ε ) 2 = ( 1 + z ¯ m 2 ) σ r , waist 2 ,
σ k ( ε ) 2 = ( 1 2 V N L ( z m ) z m ) σ k , waist 2 ,
V N L ( z m ) = P P N L ε ¯ n 0 ( σ r , waist 2 σ r , L ( z m ) 2 ) 2 1 z R .
σ r , system 2 = ( 1 + z ¯ m 2 ) σ r , waist 2 σ k , system 2 = [ 1 2 V z m + V 2 ( z m 2 + z R 2 ) ] σ k , waist 2 with M system = [ 1 0 V 1 ] [ 1 z m 0 1 ] ,
A N = 1 ε n 0 j = 0 N ( N j ) V N L ( j ε n 0 + z m ) ,
B N = z m + z n 0 ε n 0 j = 0 N ( N j ) ( z m + j ε n 0 ) V N L ( j ε n 0 + z m ) ,
A = 1 P P N L β A , β A = 0 z ¯ n 0 ( z ¯ n 0 u ) [ 1 + ( u + z ¯ m ) 2 ] 2 d u ,
B = z m + z n 0 P P N L z R β B , β B = 0 z ¯ n 0 ( z ¯ n 0 u ) ( z ¯ m + u ) [ 1 + ( u + z ¯ m ) 2 ] 2 d u ,
σ r ( z ) 2 σ r , waist 2 = 1 + ( z ¯ m + z ¯ n 0 ) 2 P P N L ( z ¯ n 0 ) 2 1 + z ¯ m 2 .
σ r ( z ) 2 σ r , L ( z m ) 2 = ( 1 + z ¯ m z ¯ n 0 1 + z ¯ m 2 ) 2 + ( z ¯ n 0 1 + z ¯ m 2 ) 2 ( 1 P P N L ) ,
R 1 = n 0 z m ( 1 + ( π w 0 2 λ z m ) 2 ) = n 0 z R z ¯ m ( 1 + z ¯ m 2 ) ,
w 1 2 = w 0 2 ( 1 + ( λ z m π w 0 2 ) 2 ) = w 0 2 ( 1 + z ¯ m 2 ) ,
σ r ( z ) 2 σ r , L ( z m ) 2 = ( 1 + z R 1 ) 2 + ( λ z π n 0 w 1 2 ) 2 ( 1 P P N L ) .
σ r ( z ) 2 σ r , L ( z m + z n 0 ) 2 z 1 P P N L 1 1 + z ¯ m 2 .
S transmittance = E ̃ N L ( k = 0 , e ) 2 E ̃ L ( k = 0 , z m ) 2 = E N L ( r , e ) d 2 r 2 E L ( r , z m ) d 2 r 2 .
S variance = σ k , N L 2 σ k , L 2 = E N L ( r , e ) E N L ( r , e ) * d 2 r E L ( r , z m ) E L ( r , z m ) * d 2 r ,
C N = j = 0 N V N L ( j ε n 0 + z m ) ,
D N = 1 j = 0 N ( z m + j ε n 0 ) V N L ( j ε n 0 + z m ) .
C = P P N L β C z R , β C = 0 e ¯ n 0 1 [ 1 + ( u + z ¯ m ) 2 ] 2 d u ,
D = 1 P P N L β D , β D = 0 e ¯ n 0 ( z ¯ m + u ) [ 1 + ( u + z ¯ m ) 2 ] 2 d u ,
S variance = 1 P P N L ( e ¯ n 0 + 2 z ¯ m ) e ¯ n 0 ( 1 + z ¯ m 2 ) [ ( e ¯ n 0 + z ¯ m ) 2 + 1 ] ,
Δ Z p v variance = ( e ¯ n 0 ) 2 4 + 2 [ ( e ¯ n 0 ) 2 + 4 ] 2 4 ( e ¯ n 0 ) 2 3 z R ,
Δ S p v variance = 24 ( e ¯ n 0 ) Δ Z p v variance z R [ ( e ¯ n 0 ) 2 + 4 ] 2 [ ( e ¯ n 0 ) 2 4 ] ( Δ Z p v variance z R ) 2 P P N L .
S variance = 1 P P N L 2 z ¯ m e ¯ n 0 ( 1 + z ¯ m 2 ) 2 .
σ r u , out 2 = ( r u ) 2 W in ( D r B k k 0 , C k 0 r + A k ) d 2 r d 2 k W out ( r , k ) d 2 r d 2 k .
σ r . u , out 2 = ( A r u + B k 0 k u ) 2 W in ( r , k ) d 2 r d 2 k W in ( r , k ) d 2 r d 2 k .
σ r . u , out 2 = A 2 σ r . u , waist 2 + B 2 k 0 2 σ k . u , waist 2 .
I ( r , z m ) = γ 2 I ( γ r , z m = 0 ) , γ = σ r , waist σ r , L ( z m ) ,
φ ( r , z m ) = ϕ ( γ r ) z ¯ m , ϕ ( r ) = k 0 2 z R r 2 .
σ k u ( ε ) 2 σ k u , waist 2 = k 0 2 n 2 ε z ¯ γ 4 2 z R P [ ( u ) I ( r , 0 ) 2 ] [ ( u ) r 2 ] d 2 r [ I ( r , 0 ) d 2 r ] 2 ,
( u ) ( I ( r , 0 ) 2 r u ) d 2 r = 0 ,
[ ( u ) I ( r , 0 ) 2 ] r u d 2 r = I ( r , 0 ) 2 [ ( u ) r u ] d 2 r ,
σ k u ( ε ) 2 σ k u , waist 2 = k 0 2 n 2 ε z ¯ γ 4 z R P I ( r , 0 ) 2 d 2 r [ I ( r , 0 ) d 2 r ] 2 .
σ r . u , waist 2 = 1 2 σ r , waist 2 ( 1 + 1 2 cos ( 2 β ) δ q 1 ) for LG p , q ,
σ r . u , waist 2 = 1 2 σ r , waist 2 ( 1 + cos ( 2 β ) m n 1 + m + n ) for HG m , n ,
β A = z ¯ m z ¯ n 0 2 ( 1 + z ¯ m 2 ) z ¯ m + z ¯ n 0 2 [ arctan ( z ¯ m ) arctan ( z ¯ m + z ¯ n 0 ) ] ,
β B = z ¯ n 0 2 ( 1 + z ¯ m 2 ) + 1 2 [ arctan ( z ¯ m ) arctan ( z ¯ m + z ¯ n 0 ) ] ,
β D = z ¯ n 0 ( z ¯ n 0 + 2 z ¯ m ) 2 ( 1 + z ¯ m 2 ) [ 1 + ( z ¯ m + z ¯ n 0 ) 2 ] .

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