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.

© 2010 OSA

1. Introduction

With the identical spatial extension, the angular spreading of a Lorentz-Gaussian distribution is higher than that of a Gaussian distribution [1]. Therefore, Lorentz-Gauss beams provide appropriate models to describe the radiation emitted by a single mode diode laser [2,3]. The Lorentz beam is a special case of Lorentz-Gauss beams. Within the framework of the paraxial and non-paraxial cases, the properties of Lorentz-Gauss beams have been extensively investigated [414]. However, the reported researches were mainly confined to the case of fully coherent Lorentz-Gauss beams. In the practical optical systems, laser beams are almost partially coherent [15], which denotes that fully coherent laser sources are the ideal cases. In the paraxial optics, an arbitrary optical system is described by an ABCD matrix, which is very simple and convenient to the practical applications. To properly design an optical system that includes a single mode diode laser, the analysis of propagation of a Lorentz-Gauss beam passing through an ABCD optical system is prerequisite. In the remainder of this paper, therefore, the propagation of a partially coherent Lorentz-Gauss beam through a paraxial and real ABCD optical system is investigated. Moreover, analytical expressions for the mutual coherence function, the effective beam size, and the spatial complex degree of coherence are derived by means of the mathematical techniques. A numerical example is also demonstrated.

2. Propagation of a partially coherent Lorentz-Gauss beam through a paraxial and real ABCD optical system

In the Cartesian coordinate system, the z-axis is taken to be the propagation axis. The mutual coherence function of a partially coherent Lorentz-Gauss beam in the source plane z=0 is characterized by

Γ(x01,x02;y01,y02;0)=E(x01,y01,0)E(x02,y02,0)g(x01,x02;y01,y02),
with E(x 01,y 01,0), E(x 02,y 02,0), and g(x 01, x 02;y 01, y 02) given by
E(x0p,y0p,0)=1w0xw0y[1+(x0p/w0x)2][1+(y0p/w0y)2]exp(x0p2+y0p2w02),
g(x01,x02;y01,y02)=exp[(x01x02)22σx2(y01y02)22σy2],
where p=1 or 2 (hereafter). w 0 x and w 0 y are the parameters related to the beam widths of the Lorentz part in the x- and y-directions, respectively. w 0 is the waist of the Gaussian part. g(x01,x02;y01,y02) is the complex degree of spatial coherence of beams generated by a Schell-model source. σx and σy are the spatial coherence length in the x- and y-directions, respectively. The time-dependent factor exp(-iωt) is omitted in the Eq. (1), and ω is the circular frequency. The Lorentz distribution can be expanded into the linear superposition of Hermite-Gaussian functions:
1(x0p2+w0x2)(y0p2+w0y2)=π2w0x2w0y2m=0Nn=0Na2ma2nH2m(x0pw0x)H2n(y0pw0y)exp(x0p2w0x2y0p2w0y2),
where N is the number of the expansion. a 2 m and a 2 n are the weight coefficients and can be indexed in [16]. H 2 m(.) and H 2 n(.) are the 2mth- and 2nth-order Hermite polynomials, respectively. Therefore, Eq. (2) can be rewritten as follows:
E(x0p,y0p,0)=π2w0xw0ym=0Nn=0Na2ma2nH2m(x0pw0x)H2n(y0pw0y)exp(x0p2wx2y0p2wy2),
where
1wj2=1w02+12w0j2,
and j=x or y (hereafter). The propagation of a partially coherent Lorentz-Gauss beam through a paraxial ABCD optical system is described by the generalized Huygens-Fresnel diffraction integral:
Γ(x1,x2;y1,y2;z)=1λBexp(ikD2B[(x22+y22)(x12+y12)])Γ(x01,x02;y01,y02;0)×exp{ik2B[A(x022+y022)A(x012+y012)2(x02x2+y02y2)+2(x01x1+y01y1)]}dx01dx02dy01dy02,
where k=2π/λ is the wave number. λ is the wavelength. A, B, C, and D are matrix elements of the optical system between the source and the output planes. Moreover, there is no inherent aperture between the source and the output planes. Therefore, A, B, C, and D are all real-valued. Using the following mathematical formulae [17]:
H2m(x)exp[(xy)2/u]dx=π/u(1u)mH2m[(1u)1/2y],
H2m(x)=l=0m(1)l(2m)!l!(2m2l)!(2x)2m2l,
x2nexp(bx2+2cx)dx=(2n)!πb(cb)2nexp(c2b)s=0n1s!(2n2s)!(b4c2)s,
we can obtain the mutual coherence function of a partially coherent Lorentz-Gauss beam in the output plane as
Γ(x1,x2;y1,y2;z)=Γ(x1,x2,z)Γ(y1,y2,z),
with Γ(x1,x2,z)and Γ(y1,y2,z) given by
Γ(j1,j2,z)=π22iλBα1jα2jexp(βj2α2jk2w0j2j224α1jB2+ikD2B(j22j12))m=0Nm=0Na2ma2m(11α1j)m×l1=0m(1)l1(2m)!l1!(2m2l1)!l2=0m(1)l2(2m)!l2!(2m2l2)!l3=02m2l2(l32m2l2)22(m+m)2l12l2γjl3δj2m2l2l3×(2m+l32l1)!(βjα2j)2m+l32l1s=0[ml1+l3/2]1s!(2m+l32l12s)!(α2j4βj2)s,
where [m-l 1+l 3/2] gives the greatest integer less than or equal to (m-l 1+l 3/2), and

α1j=(1wj2+12σj2ikA2B)w0j2,α2j=(1wj2+12σj2+ikA2B)w0j2w0j44α1jσj4,
βj=ikw0jj12Bikw0j3j24α1jBσj2,γj=w0j22(α1j2α1j)1/2σj2,δj=kw0jj22iB(α1j2α1j)1/2.

The effective beam size of the partially coherent Lorentz-Gauss beam in the x- and y-directions of the output plane is defined as [18]

Wjz=2j2Γ(x,x;y,y;z)dxdyΓ(x,x;y,y;z)dxdy.
Substituting Eq. (11) into Eq. (15), the analytical effective beam size of the partially coherent Lorentz-Gauss beam yields
Wjz=2Ω1jΩ2j,
with Ω1 j and Ω2 j given by
Ω1j=m=0Nm=0Na2ma2m(11α1j)ml1=0m(1)l1(2m)!l1!(2m2l1)!l2=0m(1)l2(2m)!l2!(2m2l2)!l3=02m2l2(l32m2l2)×22(m+m)2l12l2γjl3ξj2m2l2l3(2m+l32l1)!s=0[ml1+l3/2]1s!(2m+l32l12s)!4s×α2j2l1+s2ml3ηj2m+l32l12sΓ(m+ml1l2s+1.5)ζjl1+l2+smm1.5,
Ω2j=m=0Nm=0Na2ma2m(11α1j)ml1=0m(1)l1(2m)!l1!(2m2l1)!l2=0m(1)l2(2m)!l2!(2m2l2)!l3=02m2l2(l32m2l2)×22(m+m)2l12l2γjl3ξj2m2l2l3(2m+l32l1)!s=0[ml1+l3/2]1s!(2m+l32l12s)!4s×α2j2l1+s2ml3ηj2m+l32l12sΓ(m+ml1l2s+0.5)ζjl1+l2+smm0.5,
where
ξj=kw0j2iB(α1j2α1j)1/2,ηj=ikw0j2Bikw0j34α1jBσj2,ζj=k2w0j24α1jB2ηj2α2j,
and Γ(.) is a Gamma function.

The spatial complex degree of coherence of the partially coherent Lorentz-Gauss beam at two points (x 1, y 1, z) and (x 2, y 2, z) turns out to [19,20]

μ(x1,x2;y1,y2;z)=μ(x1,x2,z)μ(y1,y2,z),
with μ(x1,x2,z)and μ(y1,y2,z)given by
μ(j1,j2,z)=Γ(j1,j2,z)[Γ(j1,j1,z)Γ(j2,j2,z)]1/2.
Inserting Eq. (12) into Eq. (21), one can calculate the spatial complex degree of coherence.

3. Numerical calculations and analyses

Now, we consider the focusing of the partially coherent Lorentz-Gauss beam. A thin lens with the focal length f is placed in front of the single mode diode laser, so that the partially coherent Lorentz-Gauss beam is transformed into a converging beam. In the case of diode-fiber coupling, a fiber end is placed in the focal region. The matrix elements of this optical arrangement are A=0, B=f, C=−1/f, and D=1. The normalized light intensity, the effective beam size, and the spatial complex degree of coherence for the partially coherent Lorentz-Gauss beam in the focal plane are calculated by using the formulae derived above. As the x- and y-directions are separable, only the x-direction is considered in the following calculations. Moreover, we mainly concentrate on the effect of the spatial coherence length. Figure 1 represents the normalized intensity distribution in the focal plane. The parameters used are chosen as follow: λ=0.8μm, f=1m and w 0 x=1mm. w 0=2mm in Fig. 1(a) and w 0=∞ in Fig. 1(b). The normalized intensity distribution of the partially coherent Lorentz-Gauss beam spreads with decreasing the spatial coherence length σx. The effective beam size of partially coherent Lorentz-Gauss beams in the focal plane versus the coherence length σx is depicted in Fig. 2 . The inset shows the detail change of Wxz within the range of 0.2mm≤σx≤1mm. With increasing the spatial coherence length σx, the effective beam size first quickly decreases and then tends to a minimum value. The better coherence the partially coherent Lorentz-Gauss beam has, the smaller effective beam size it has. If the spatial coherence length σx keeps invariant, the partially coherent Lorentz-Gauss beam with smaller w 0 and w 0 x has the larger effective beam size. Figure 3 shows the spatial complex degree of coherence in the focal plane. w 0=2mm and w 0 x=1mm. σx=1mm, 2mm, and 210mm in Figs. 3(a)3(c), respectively. With given two given points (x 1, f) and (x 2, f), their spatial complex degree of coherence increases with increasing the spatial coherence length σx. To quantitatively evaluate the influence of the spatial coherence length on the spatial complex degree of coherence, we calculate the spatial complex degree of coherence at two points (0.1mm, f)and (0.4mm, f) by altering the spatial coherence length σx, which is shown in Fig. 4 . With increasing the spatial coherence length σx, the spatial complex degree of coherence first quickly increases and then tends to the saturated value 1. With a given σx, the partially coherent Lorentz-Gauss beam with the smaller w 0 and w 0 x has the larger spatial complex degree of coherence.

 

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=∞.

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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.

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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=210mm.

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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.

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It should be pointed out that the presented formulae are valid for the case of the three beam parameters being far larger than the wavelength. In the paraxial case, the beam propagation fa-ctor of a partially coherent Lorentz-Gauss beam has been presented as Eq. (22) in [21]. By expanding the complementary error function, the beam propagation factor of a partially coherent Lorentz-Gauss beam can be verified to be larger than that of a corresponding partially coherent Gaussian beam. Therefore, the angular spreading of a partially coherent Lorentz-Gauss beam still preserves high within the paraxial propagation. When the three beam parameters are of the order of the wavelength, an efficaciously non-paraxial method is to be sought to treat the propagation of partially coherent Lorentz-Gauss beams.

4. Conclusions

Based on the generalized Huygens-Fresnel integral and the expansion of Lorentz distribution, the analytical propagation equation of a partially coherent Lorentz-Gauss beam through a paraxial and real ABCD optical system is derived. Moreover, the analytical formulae for the effective beam size and the spatial complex degree of coherence are also presented. As a numerical example, the normalized light intensity, the effective beam size, and the spatial complex degree of coherence for the partially coherent Lorentz-Gauss beam focused by a thin lens are calculated in the focal plane. The effect of the spatial coherence length is mainly discussed. With increasing the spatial coherence length, the partially coherent Lorentz-Gauss beam has the smaller effective beam size and the higher spatial complex degree of coherence. As apertures usually exist in the practical optical system, the propagation of a partially coherent Lorentz-Gauss beam through a paraxial and complex ABCD optical system also deserves to be investigated. This research is useful to the optical designs that are involved in the single mode diode laser.

Acknowledgements

This research was supported by National Natural Science Foundation of China under Grant No. 10974179 and Zhejiang Provincial Natural Science Foundation of China under Grant No. Y1090073.

References and links

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). [CrossRef]   [PubMed]  

19. A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5(5), 713–720 (1988). [CrossRef]  

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]  

References

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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).
    [CrossRef] [PubMed]
  19. A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5(5), 713–720 (1988).
    [CrossRef]
  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)

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]

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]

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

G. Zhou, “Fractional Fourier transform of Lorentz-Gauss beams,” J. Opt. Soc. Am. A 26(2), 350–355 (2009).
[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, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96(1), 149–153 (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]

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[CrossRef]

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[CrossRef]

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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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