Abstract

The q-parameter of a Gaussian beam is a convenient way to determine its paraxial propagation in a medium as well as in an optical system under external or induced lensing. The assumption is that the Gaussian beam either is scalar or has a linear polarization. It is shown that propagation of radially polarized Gaussian beams in a medium and/or under lensing can be readily analyzed rather simply by knowing the q-transformation of the underlying scalar Gaussian beam. The exact profiles of the longitudinal and transverse components of initially radially polarized lowest-order Laguerre–Gaussian beams are derived and compared with those of the linearly polarized Gaussian beam. It can be readily shown that the longitudinal component of the polarization does not contribute to real power flow at the focal plane. The focal shift and the Guoy phase during lensing are calculated, again based on the underlying q-parameter. The methodology for extension to higher-order Laguerre–Gaussians is also developed. Finally, waveguiding of radially polarized beams in a graded index square law medium is analyzed, and conditions for the existence of radially or longitudinally polarized modes are derived.

© 2009 Optical Society of America

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References

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2007 (4)

2006 (1)

2005 (2)

2004 (2)

R. Borghi, and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029-2037 (2004).
[CrossRef]

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

2003 (1)

2002 (2)

A. Ciattoni, B. Crosignami, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17-20 (2002).
[CrossRef]

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B B74, S83-S88 (2002).
[CrossRef]

2001 (2)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating,” Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

1999 (2)

1998 (1)

1974 (1)

G. N. Vinokurov, A. A. Mak, and V. M. Mitkin, “Generating azimuthal and radial modes in optical resonators,” Kvant. Elektron. (Kiev) 8, 1890-1891 (1974) (in Russian).

Aït-Ameur, K.

Armstrong, D. J.

Banerjee, P. P.

T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing using MATLAB (Elsevier, 2001).

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Bomzon, Z.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating,” Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

Borghi, R.

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Chen, H.

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259-261 (2007).
[CrossRef]

Ciattoni, A.

A. Ciattoni, B. Crosignami, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17-20 (2002).
[CrossRef]

Crosignami, B.

A. Ciattoni, B. Crosignami, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17-20 (2002).
[CrossRef]

Deng, D.

Denis, R. de S.

Dong, J.

Gahagan, K. T.

Glur, H.

Guo, Q.

Hasman, E.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating,” Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

Hayazawa, N.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

Hierle, R.

Kawata, S.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

Kleiner, V.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating,” Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

Li, Y.-P.

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259-261 (2007).
[CrossRef]

Lou, Q.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1966).

Mak, A. A.

G. N. Vinokurov, A. A. Mak, and V. M. Mitkin, “Generating azimuthal and radial modes in optical resonators,” Kvant. Elektron. (Kiev) 8, 1890-1891 (1974) (in Russian).

Mitkin, V. M.

G. N. Vinokurov, A. A. Mak, and V. M. Mitkin, “Generating azimuthal and radial modes in optical resonators,” Kvant. Elektron. (Kiev) 8, 1890-1891 (1974) (in Russian).

Nesterov, A. V.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

Niziev, V. G.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Passilly, N.

Philips, M. C.

Piché, M.

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B B74, S83-S88 (2002).
[CrossRef]

Poon, T.-C.

T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing using MATLAB (Elsevier, 2001).

Porto, P. D.

A. Ciattoni, B. Crosignami, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17-20 (2002).
[CrossRef]

Roch, J.-F.

Roth, M. S.

Saito, Y.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

Santarsiero, M.

Shih, C.-C.

C.-C. Shih, “Radial polarization laser resonator,” U.S. patent 5,359,622, October 25, 1994.

Smith, A. V.

Swartzlander, G. A.

Tovar, A. A.

Treussart, F.

Varin, C.

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B B74, S83-S88 (2002).
[CrossRef]

Vinokurov, G. N.

G. N. Vinokurov, A. A. Mak, and V. M. Mitkin, “Generating azimuthal and radial modes in optical resonators,” Kvant. Elektron. (Kiev) 8, 1890-1891 (1974) (in Russian).

Weber, H. P.

Wei, Y.

Wu, G.

Wu, L.

Wyss, E. W.

Yang, X.

Yariv, A.

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford Univ. Press, 2007).

Yeh, P.

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford Univ. Press, 2007).

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Zhan, Q.

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259-261 (2007).
[CrossRef]

Zhang, Y.

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259-261 (2007).
[CrossRef]

Zhou, J.

Appl. Opt. (2)

Appl. Phys. B (1)

C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity TM01 laser beams,” Appl. Phys. B B74, S83-S88 (2002).
[CrossRef]

Appl. Phys. Lett. (2)

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using spacevariant subwavelength metal strip grating,” Appl. Phys. Lett. 79, 1587-1589 (2001).
[CrossRef]

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85, 6239-6241 (2004).
[CrossRef]

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

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

J. Phys. D (1)

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455-1461 (1999).
[CrossRef]

Kvant. Elektron. (Kiev) (1)

G. N. Vinokurov, A. A. Mak, and V. M. Mitkin, “Generating azimuthal and radial modes in optical resonators,” Kvant. Elektron. (Kiev) 8, 1890-1891 (1974) (in Russian).

Opt. Commun. (1)

A. Ciattoni, B. Crosignami, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17-20 (2002).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

H. Chen, Q. Zhan, Y. Zhang, and Y.-P. Li, “The Gouy phase shift of the highly focused radially polarized beam,” Phys. Lett. A 371, 259-261 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Other (4)

C.-C. Shih, “Radial polarization laser resonator,” U.S. patent 5,359,622, October 25, 1994.

T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing using MATLAB (Elsevier, 2001).

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford Univ. Press, 2007).

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, 1966).

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Figures (4)

Fig. 1
Fig. 1

Intensity profiles of transversely and radially polarized Gaussian beams at the back focal plane of a lens of focal length f = 5 cm . The initial intensity waist sizes of both beams are taken as d 0 I = 2 w 0 I = 2 w 0 = 2.5 cm . Also, E 0 = E g 0 , with E g 0 corresponding to a transversely polarized Gaussian beam power of P g 0 = 10 mW . The sequence of figures from top to bottom are the intensity profiles of the transversely (x-) polarized Gaussian beam; the x-, y-, and z-components of the radially polarized Gaussian beam for θ = 45 ; and the z-component of the transversely (x-) polarized Gaussian beam, respectively.

Fig. 2
Fig. 2

Total intensity profiles of radially polarized Gaussian beams at the back focal plane of a lens of focal length f = 5 cm . The initial intensity waist sizes of the beams are taken as d 0 I = 2.5 cm (top figure) and d 0 I = 5 cm (bottom figure), corresponding to f-numbers of 2 and 1, respectively. All other parameters are identical to those in Fig. 1.

Fig. 3
Fig. 3

Intensity profiles of transversely and radially polarized Gaussian beams monitored at a distance z = f ± 0.01 f behind a lens of focal length f = 5 cm . All beam parameters and the sequence of figures are the same as in Fig. 1.

Fig. 4
Fig. 4

Top: On-axis intensity of z-polarized component of focused radially polarized Gaussian beam at the back focal plane as a function of the f-number. Bottom: On-axis intensity of x-component of focused linearly (x-) polarized Gaussian beam at the back focal plane, also as a function of the f-number, shown for comparison. The parameters of the Gaussian beams are identical to that in Fig. 1.

Equations (59)

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E p ( x , y ; z ) = E p ( x , y ; z = 0 ) h ( x , y ; z ) ,
h ( x , y ; z ) = j k 0 2 π z e j k 0 z exp [ j k 0 ( x 2 + y 2 ) 2 z ] , j = 1 ,
E ̃ p ( k x , k y ; z ) = E ̃ p ( k x , k y ; z = 0 ) h ̃ ( k x , k y ; z ) ,
h ̃ ( k x , k y ; z ) = e j k 0 z exp [ j ( k x 2 + k y 2 ) z 2 k 0 ]
f ̃ ( k x , k y ) = I x , y { f ( x , y ) } = f ( x , y ) exp [ j ( k x x + k y y ) ] d x d y ,
f ( x , y ) = I x , y 1 { f ̃ ( k x , k y ) } = ( 1 4 π 2 ) f ̃ ( k x , k y ) exp [ j ( k x x + k y y ) ] d k x d k y .
E p g ( x , y ; 0 ) = E g 0 exp [ α ( x 2 + y 2 ) ] ,
E ̃ p g ( k x , k y ; 0 ) = E g 0 ( π α ) exp [ ( k x 2 + k y 2 ) 4 α ] ,
E ̃ p g ( k x , k y ; z ) = E ̃ p g ( k x , k y ; 0 ) h ̃ ( k x , k y ; z ) = E g 0 ( π α ) e j k 0 z exp [ ( k x 2 + k y 2 ) 4 α ] × exp [ j ( k x 2 + k y 2 ) z 2 k 0 ] E g 0 ( π α ) e j k 0 z exp [ j ( k x 2 + k y 2 ) q 2 k 0 ] ,
E p g ( x , y ; z ) = ( j k 0 2 π q ) E g 0 ( π α ) e j k 0 z exp [ j k 0 ( x 2 + y 2 ) 2 q ] .
q = z + j k 0 2 α .
α = 1 w 0 2 j k 0 2 f = ( 1 w 0 2 ) ( 1 j z R f ) .
q = z + j z R 1 j ( z R f ) = ( z z R 2 f 1 + ( z R f ) 2 ) + j ( z R 1 + ( z R f ) 2 ) .
z = z f = f [ z R 2 ( f 2 + z R 2 ) ] .
E p x , y ( x , y ; z ) = j k 0 2 π z e j k 0 z E p x , y ( x , y ; 0 ) exp ( j k 0 ( x x ) 2 + ( y y ) 2 2 z ) d x d y ,
E p x , y ( x , y ; z ) = E p x , y ( x , y ; 0 ) h ( x , y ; z ) ,
E ̃ p x , y ( k x , k y ; z ) = E ̃ p x , y ( k x , k y ; 0 ) h ̃ ( k x , k y ; z ) .
E p x , y ( x , y ; z ) = 1 2 π E p x , y ( x , y ; 0 ) G ( R ¯ , R ¯ ) z d x d y ,
G ( R ¯ , R ¯ ) exp [ j k 0 ( R ¯ R ¯ ) ] ( R ¯ R ¯ ) e j k 0 z z exp ( j k 0 ( x x ) 2 + ( y y ) 2 2 z ) = j ( k 0 2 π ) h ( x x , y y ; z ) .
E p z ( x , y ; z ) = 1 2 π ( E p x ( x , y ; 0 ) G ( R ¯ , R ¯ ) x + E p y ( x , y ; 0 ) G ( R ¯ , R ¯ ) y ) d x d y ,
E p z ( x , y ; z ) = j ( 1 k 0 ) ( E p x ( x , y ; 0 ) ( h ( x , y ; z ) x ) + E p y ( x , y ; 0 ) ( h ( x , y ; z ) y ) ) ,
E ̃ p z ( k x , k y ; z ) = ( 1 k 0 ) [ k x E ̃ p x ( k x , k y ; 0 ) + k y E ̃ p y ( k x , k y ; 0 ) ] h ̃ ( k x , k y ; z ) .
I x ( y ) { f ( x , y ) x ( y ) } = j k x ( y ) f ̃ ( k x , k y ) ,
E p x ( x , y ; 0 ) = E 0 ( 2 w 0 ) x exp [ ( x 2 + y 2 ) w 0 2 ] ,
E p y ( x , y ; 0 ) = E 0 ( 2 w 0 ) y exp [ ( x 2 + y 2 ) w 0 2 ] .
E p x ( x , y ; 0 ) = E 0 ( 2 w 0 ) x exp [ α ( x 2 + y 2 ) ] ,
E p y ( x , y ; 0 ) = E 0 ( 2 w 0 ) y exp [ α ( x 2 + y 2 ) ] ,
E p x ( y ) ( x , y ; 0 ) = x ( y ) { E 0 2 α w 0 exp [ α ( x 2 + y 2 ) ] } ,
E ̃ p x ( y ) ( k x , k y ; 0 ) = j k x ( y ) E 0 ( π 2 α 2 w 0 ) exp [ ( k x 2 + k y 2 ) 4 α ] ,
E ̃ p x ( y ) ( k x , k y ; z ) = j k x ( y ) E 0 ( π 2 α 2 w 0 ) e j k 0 z exp [ ( k x 2 + k y 2 ) 4 α ] exp [ j ( k x 2 + k y 2 ) z 2 k 0 ] = j ( k x ( y ) k 0 ) E r g 0 ( π α ) e j k 0 z exp [ j ( k x 2 + k y 2 ) q 2 k 0 ] ,
E r g 0 = E 0 ( k 0 2 α w 0 ) .
E p x ( y ) ( x , y ; z ) = j 2 π q E r g 0 ( π α ) e j k 0 z x ( y ) ( exp [ j k 0 ( x 2 + y 2 ) 2 q ] ) ,
E p x ( x , y ; z ) = k 0 2 2 2 q 2 α 2 w 0 E 0 e j k 0 z x exp [ j k 0 ( x 2 + y 2 ) 2 q ] ,
E p y ( x , y ; z ) = k 0 2 2 2 q 2 α 2 w 0 E 0 e j k 0 z y exp [ j k 0 ( x 2 + y 2 ) 2 q ] .
E p x ( x , y ; z ) E p g ( x , y ; z ) = j k 0 2 q α w 0 x E 0 E g 0 ; E p y ( x , y ; z ) E p ( x , y ; z ) = j k 0 2 q α w 0 y E 0 E g 0 ,
E ̃ p z ( k x , k y ; z ) = j E 0 ( π 2 k 0 α 2 w 0 ) ( k x 2 + k y 2 ) e j k 0 z × exp [ ( k x 2 + k y 2 ) 4 α ] exp [ j ( k x 2 + k y 2 ) z 2 k 0 ] = j E 0 ( π 2 k 0 α 2 w 0 ) ( k x 2 + k y 2 ) e j k 0 z exp [ j ( k x 2 + k y 2 ) q 2 k 0 ] = j E r g 0 ( π α ) [ ( k x 2 + k y 2 ) k 0 2 ] e j k 0 z exp [ j ( k x 2 + k y 2 ) q 2 k 0 ] ,
I x ( y ) { 2 f ( x , y ) x 2 ( y 2 ) } = k x ( y ) 2 f ̃ ( k x , k y ) ,
E p z ( x , y ; z ) = 1 2 π k 0 q E r g 0 ( π α ) e j k 0 z [ ( 2 x 2 + 2 y 2 ) ( exp [ j k 0 ( x 2 + y 2 ) 2 q ] ) ] ,
E p z ( x , y ; z ) = j k 0 2 q 2 α 2 w 0 E 0 e j k 0 z [ 1 j ( k 0 2 q ) ( x 2 + y 2 ) ] exp [ j k 0 ( x 2 + y 2 ) 2 q ] .
E p z ( x , y ; z ) E p g ( x , y ; z ) = 2 q α w 0 [ 1 j ( k 0 2 q ) ( x 2 + y 2 ) ] E 0 E g 0 .
E p z ( x , y ; z ) E p x ( x , y ; z ) = j 2 1 j ( k 0 2 q ) ( x 2 + y 2 ) k 0 x ;
E p z ( x , y ; z ) E p y ( x , y ; z ) = j 2 1 j ( k 0 2 q ) ( x 2 + y 2 ) k 0 y .
E p z g ( x , y ; z ) = j k 0 2 α q 2 x E g 0 e j k 0 z exp [ j k 0 ( x 2 + y 2 ) 2 q ] ,
[ E p z g ( x , y ; z ) E p g ] = x q .
E p z ( x , y ; z ) = j ( k 0 2 q 2 α 2 w 0 ) E 0 e j k 0 z [ 1 ( k 0 2 q ) ( x 2 + y 2 ) ] × exp [ k 0 ( x 2 + y 2 ) 2 q ] .
H ¯ p = k ¯ × E ¯ p ω 0 μ 0 k 0 ω 0 μ 0 [ E p x a ̂ y E p y a ̂ x ] .
S ¯ av = 1 2 E ¯ p × H ¯ p * = k 0 2 ω 0 μ 0 [ ( E p x 2 + E p y 2 ) a ̂ z ( E p z E p x * a ̂ x + E p z E p y * a ̂ y ) ] .
x 3 exp [ α ( x 2 + y 2 ) ] = 1 8 α 3 3 x 3 { exp [ α ( x 2 + y 2 ) ] } 3 4 α 2 x { exp [ α ( x 2 + y 2 ) ] } .
j π α [ 1 8 α 3 k x 3 + 3 4 α 2 k x ] exp [ ( k x 2 + k y 2 ) 4 α ] ,
x n e x 2 = n 1 2 x n 2 e x 2 1 2 x ( x n 1 e x 2 ) .
E p g ̱ diffr ( x , y ; Δ z ) = j n 0 k 0 2 ( q 0 + Δ z ) α 0 E g 0 e j n 0 k 0 Δ z exp [ j n 0 k 0 ( x 2 + y 2 ) 2 ( q 0 + Δ z ) ] j n 0 k 0 2 q 0 α 0 ( 1 Δ z q 0 ) E g 0 e j n 0 k 0 Δ z exp [ j n 0 k 0 ( x 2 + y 2 ) 2 q 0 ] exp [ j n 0 k 0 ( x 2 + y 2 ) Δ z 2 q 0 2 ] ,
E p g ̱ diffr ( x , y ; Δ z ) E p g ( x , y ; 0 ) ( 1 Δ z q 0 ) e j n 0 k 0 Δ z exp [ j n 0 k 0 ( x 2 + y 2 ) Δ z 2 q 0 2 ] = ( 1 + j Δ z q 0 ) e j n 0 k 0 Δ z exp [ j n 0 k 0 ( x 2 + y 2 ) Δ z 2 q 0 2 ] ,
q 0 = j q 0 ; q 0 = n 0 k 0 w 0 2 2 = n 0 k 0 2 α 0 .
e j n k 0 Δ z = e j n 0 k 0 Δ z exp [ j k 0 ( n 2 2 ) ( x 2 + y 2 ) Δ z ] e j n 0 k 0 Δ z exp [ j n 0 k 0 ( x 2 + y 2 ) 2 f ] ,
f = n 0 n 2 Δ z .
E p g ̱ lens ( x , y ; Δ z ) E p g ( x , y ; 0 ) = e j n 0 k 0 Δ z exp [ j k 0 ( n 2 2 ) ( x 2 + y 2 ) Δ z ] .
w 0 ̱ guided = 4 n 0 n 2 k 0 2 4 ,
E p z ̱ diffr ( x , y ; Δ z ) E p z ( x , y ; 0 ) e j n 0 k 0 Δ z exp [ j n 0 k 0 ( x 2 + y 2 ) Δ z q 0 2 ] .
w 0 ̱ guided ( z ) = 2 4 w 0 ̱ guided

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