Abstract

Based on the generalized Huygens-Fresnel integral and the Hermite-Gaussian expansion of a Lorentz distribution, analytical expressions for the mutual coherence function, the effective beam size, and the spatial complex degree of coherence of a partially coherent Lorentz-Gauss beam through a paraxial and real ABCD optical system are derived, respectively. As a numerical example, the focusing of a partially coherent Lorentz-Gauss beam is considered. The normalized intensity distribution, the effective beam size, and the spatial complex degree of coherence for the focused partially coherent Lorentz-Gauss beam are numerically demonstrated in the focal plane. The influence of the spatial coherence length on the normalized intensity distribution, the effective beam size, and the spatial complex degree of coherence is mainly discussed.

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  1. W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11(7), 400–402 (1975).
    [CrossRef]
  2. A. Naqwi and F. Durst, “Focus of diode laser beams: a simple mathematical model,” Appl. Opt. 29(12), 1780–1785 (1990).
    [CrossRef] [PubMed]
  3. J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
    [CrossRef]
  4. O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006).
    [CrossRef]
  5. O. Elgawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269(2), 274–284 (2007).
    [CrossRef]
  6. A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008).
    [CrossRef]
  7. G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
    [CrossRef]
  8. G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93(4), 891–899 (2008).
    [CrossRef]
  9. G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55(21), 3573–3577 (2008).
    [CrossRef]
  10. G. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25(10), 2594–2599 (2008).
    [CrossRef]
  11. G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26(1), 141–147 (2009).
    [CrossRef]
  12. G. Zhou, “Fractional Fourier transform of Lorentz-Gauss beams,” J. Opt. Soc. Am. A 26(2), 350–355 (2009).
    [CrossRef]
  13. G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41(8), 953–955 (2009).
    [CrossRef]
  14. G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96(1), 149–153 (2009).
    [CrossRef]
  15. L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
  16. P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976).
    [CrossRef]
  17. I. S. Gradshteyn, and I. M. Ryzhik, Table of integrals, series, and products (Academic Press, New York, 1980).
  18. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009).
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  19. A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5(5), 713–720 (1988).
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  20. M. Born, and E. Wolf, Principles of Optics 7th ed. (Cambridge University Press, Cambridge, UK, 1999).
  21. G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. 12(1), 015701 (2010).
    [CrossRef]

2010 (1)

G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. 12(1), 015701 (2010).
[CrossRef]

2009 (5)

2008 (6)

A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008).
[CrossRef]

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93(4), 891–899 (2008).
[CrossRef]

G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55(21), 3573–3577 (2008).
[CrossRef]

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

G. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25(10), 2594–2599 (2008).
[CrossRef]

2007 (1)

O. Elgawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269(2), 274–284 (2007).
[CrossRef]

2006 (1)

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006).
[CrossRef]

1990 (1)

1988 (1)

1976 (1)

P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976).
[CrossRef]

1975 (1)

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11(7), 400–402 (1975).
[CrossRef]

Baykal, Y.

Cai, Y.

Chen, T.

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Ding, G.

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Dumke, W. P.

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11(7), 400–402 (1975).
[CrossRef]

Durst, F.

Elgawhary, O.

O. Elgawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269(2), 274–284 (2007).
[CrossRef]

Evans, W. A. B.

A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008).
[CrossRef]

Eyyuboglu, H. T.

Friberg, A. T.

Gawhary, O. E.

A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008).
[CrossRef]

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006).
[CrossRef]

Naqwi, A.

Qu, J.

Schmidt, P. P.

P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976).
[CrossRef]

Severini, S.

A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008).
[CrossRef]

O. Elgawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269(2), 274–284 (2007).
[CrossRef]

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006).
[CrossRef]

Torre, A.

A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008).
[CrossRef]

Turunen, J.

Xu, Y.

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

Yang, J.

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Yuan, X.

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Yuan, Y.

Zheng, J.

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

Zhou, G.

G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. 12(1), 015701 (2010).
[CrossRef]

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41(8), 953–955 (2009).
[CrossRef]

G. Zhou, “Fractional Fourier transform of Lorentz-Gauss beams,” J. Opt. Soc. Am. A 26(2), 350–355 (2009).
[CrossRef]

G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96(1), 149–153 (2009).
[CrossRef]

G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26(1), 141–147 (2009).
[CrossRef]

G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93(4), 891–899 (2008).
[CrossRef]

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55(21), 3573–3577 (2008).
[CrossRef]

G. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25(10), 2594–2599 (2008).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (2)

G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93(4), 891–899 (2008).
[CrossRef]

G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96(1), 149–153 (2009).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11(7), 400–402 (1975).
[CrossRef]

J. Mod. Opt. (2)

G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55(21), 3573–3577 (2008).
[CrossRef]

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

J. Opt. (1)

G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. 12(1), 015701 (2010).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006).
[CrossRef]

A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008).
[CrossRef]

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

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

J. Phys. B (1)

P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976).
[CrossRef]

Opt. Commun. (1)

O. Elgawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269(2), 274–284 (2007).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41(8), 953–955 (2009).
[CrossRef]

Proc. SPIE (1)

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Other (3)

L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).

I. S. Gradshteyn, and I. M. Ryzhik, Table of integrals, series, and products (Academic Press, New York, 1980).

M. Born, and E. Wolf, Principles of Optics 7th ed. (Cambridge University Press, Cambridge, UK, 1999).

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Figures (4)

Fig. 1
Fig. 1

Normalized intensity distribution in the x-direction of partially coherent Lorentz-Gauss beams with different σx in the focal plane. w 0 x =1mm. (a) w 0=2mm. (b) w 0=∞.

Fig. 2
Fig. 2

The effective beam size in the x-direction of partially coherent Lorentz-Gauss beams in the focal plane versus the coherence length σx . (a) w 0=2mm. (b) w 0 x =2mm.

Fig. 3
Fig. 3

The spatial complex degree of coherence in the x-direction of partially coherent Lorentz-Gauss beams in the focal plane. w 0=2mm and w 0 x =1mm. (a) σx =1mm. (b) σx =2mm. (c) σx = 2 10 mm.

Fig. 4
Fig. 4

The spatial complex degree of coherence in the x-direction of partially coherent Lorentz-Gauss beams in the focal plane versus the coherence length σx . x 1=0.1mm and x 2=0.4mm. (a) w 0=2mm. (b) w 0 x =2mm.

Equations (21)

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Γ ( x 01 , x 02 ; y 01 , y 02 ; 0 ) = E ( x 01 , y 01 , 0 ) E ( x 02 , y 02 , 0 ) g ( x 01 , x 02 ; y 01 , y 02 ) ,
E ( x 0 p , y 0 p , 0 ) = 1 w 0 x w 0 y [ 1 + ( x 0 p / w 0 x ) 2 ] [ 1 + ( y 0 p / w 0 y ) 2 ] exp ( x 0 p 2 + y 0 p 2 w 0 2 ) ,
g ( x 01 , x 02 ; y 01 , y 02 ) = exp [ ( x 01 x 02 ) 2 2 σ x 2 ( y 01 y 02 ) 2 2 σ y 2 ] ,
1 ( x 0 p 2 + w 0 x 2 ) ( y 0 p 2 + w 0 y 2 ) = π 2 w 0 x 2 w 0 y 2 m = 0 N n = 0 N a 2 m a 2 n H 2 m ( x 0 p w 0 x ) H 2 n ( y 0 p w 0 y ) exp ( x 0 p 2 w 0 x 2 y 0 p 2 w 0 y 2 ) ,
E ( x 0 p , y 0 p , 0 ) = π 2 w 0 x w 0 y m = 0 N n = 0 N a 2 m a 2 n H 2 m ( x 0 p w 0 x ) H 2 n ( y 0 p w 0 y ) exp ( x 0 p 2 w x 2 y 0 p 2 w y 2 ) ,
1 w j 2 = 1 w 0 2 + 1 2 w 0 j 2 ,
Γ ( x 1 , x 2 ; y 1 , y 2 ; z ) = 1 λ B exp ( i k D 2 B [ ( x 2 2 + y 2 2 ) ( x 1 2 + y 1 2 ) ] ) Γ ( x 01 , x 02 ; y 01 , y 02 ; 0 ) × exp { i k 2 B [ A ( x 02 2 + y 02 2 ) A ( x 01 2 + y 01 2 ) 2 ( x 02 x 2 + y 02 y 2 ) + 2 ( x 01 x 1 + y 01 y 1 ) ] } d x 01 d x 02 d y 01 d y 02 ,
H 2 m ( x ) exp [ ( x y ) 2 / u ] d x = π / u ( 1 u ) m H 2 m [ ( 1 u ) 1 / 2 y ] ,
H 2 m ( x ) = l = 0 m ( 1 ) l ( 2 m ) ! l ! ( 2 m 2 l ) ! ( 2 x ) 2 m 2 l ,
x 2 n exp ( b x 2 + 2 c x ) d x = ( 2 n ) ! π b ( c b ) 2 n exp ( c 2 b ) s = 0 n 1 s ! ( 2 n 2 s ) ! ( b 4 c 2 ) s ,
Γ ( x 1 , x 2 ; y 1 , y 2 ; z ) = Γ ( x 1 , x 2 , z ) Γ ( y 1 , y 2 , z ) ,
Γ ( j 1 , j 2 , z ) = π 2 2 i λ B α 1 j α 2 j exp ( β j 2 α 2 j k 2 w 0 j 2 j 2 2 4 α 1 j B 2 + i k D 2 B ( j 2 2 j 1 2 ) ) m = 0 N m = 0 N a 2 m a 2 m ( 1 1 α 1 j ) m × l 1 = 0 m ( 1 ) l 1 ( 2 m ) ! l 1 ! ( 2 m 2 l 1 ) ! l 2 = 0 m ( 1 ) l 2 ( 2 m ) ! l 2 ! ( 2 m 2 l 2 ) ! l 3 = 0 2 m 2 l 2 ( l 3 2 m 2 l 2 ) 2 2 ( m + m ) 2 l 1 2 l 2 γ j l 3 δ j 2 m 2 l 2 l 3 × ( 2 m + l 3 2 l 1 ) ! ( β j α 2 j ) 2 m + l 3 2 l 1 s = 0 [ m l 1 + l 3 / 2 ] 1 s ! ( 2 m + l 3 2 l 1 2 s ) ! ( α 2 j 4 β j 2 ) s ,
α 1 j = ( 1 w j 2 + 1 2 σ j 2 i k A 2 B ) w 0 j 2 , α 2 j = ( 1 w j 2 + 1 2 σ j 2 + i k A 2 B ) w 0 j 2 w 0 j 4 4 α 1 j σ j 4 ,
β j = i k w 0 j j 1 2 B i k w 0 j 3 j 2 4 α 1 j B σ j 2 , γ j = w 0 j 2 2 ( α 1 j 2 α 1 j ) 1 / 2 σ j 2 , δ j = k w 0 j j 2 2 i B ( α 1 j 2 α 1 j ) 1 / 2 .
W j z = 2 j 2 Γ ( x , x ; y , y ; z ) d x d y Γ ( x , x ; y , y ; z ) d x d y .
W j z = 2 Ω 1 j Ω 2 j ,
Ω 1 j = m = 0 N m = 0 N a 2 m a 2 m ( 1 1 α 1 j ) m l 1 = 0 m ( 1 ) l 1 ( 2 m ) ! l 1 ! ( 2 m 2 l 1 ) ! l 2 = 0 m ( 1 ) l 2 ( 2 m ) ! l 2 ! ( 2 m 2 l 2 ) ! l 3 = 0 2 m 2 l 2 ( l 3 2 m 2 l 2 ) × 2 2 ( m + m ) 2 l 1 2 l 2 γ j l 3 ξ j 2 m 2 l 2 l 3 ( 2 m + l 3 2 l 1 ) ! s = 0 [ m l 1 + l 3 / 2 ] 1 s ! ( 2 m + l 3 2 l 1 2 s ) ! 4 s × α 2 j 2 l 1 + s 2 m l 3 η j 2 m + l 3 2 l 1 2 s Γ ( m + m l 1 l 2 s + 1.5 ) ζ j l 1 + l 2 + s m m 1.5 ,
Ω 2 j = m = 0 N m = 0 N a 2 m a 2 m ( 1 1 α 1 j ) m l 1 = 0 m ( 1 ) l 1 ( 2 m ) ! l 1 ! ( 2 m 2 l 1 ) ! l 2 = 0 m ( 1 ) l 2 ( 2 m ) ! l 2 ! ( 2 m 2 l 2 ) ! l 3 = 0 2 m 2 l 2 ( l 3 2 m 2 l 2 ) × 2 2 ( m + m ) 2 l 1 2 l 2 γ j l 3 ξ j 2 m 2 l 2 l 3 ( 2 m + l 3 2 l 1 ) ! s = 0 [ m l 1 + l 3 / 2 ] 1 s ! ( 2 m + l 3 2 l 1 2 s ) ! 4 s × α 2 j 2 l 1 + s 2 m l 3 η j 2 m + l 3 2 l 1 2 s Γ ( m + m l 1 l 2 s + 0.5 ) ζ j l 1 + l 2 + s m m 0.5 ,
ξ j = k w 0 j 2 i B ( α 1 j 2 α 1 j ) 1 / 2 , η j = i k w 0 j 2 B i k w 0 j 3 4 α 1 j B σ j 2 , ζ j = k 2 w 0 j 2 4 α 1 j B 2 η j 2 α 2 j ,
μ ( x 1 , x 2 ; y 1 , y 2 ; z ) = μ ( x 1 , x 2 , z ) μ ( y 1 , y 2 , z ) ,
μ ( j 1 , j 2 , z ) = Γ ( j 1 , j 2 , z ) [ Γ ( j 1 , j 1 , z ) Γ ( j 2 , j 2 , z ) ] 1 / 2 .

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